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AN 



ELEMENTARY TREATISE 



ON THE 



DIFFERENTIAL CALCULUS 

FOUNDED ON THE 

. •- 

METHOD OF RATES OR FLUXIONS 



BY • 

JOHN MINOT RICE 

PROFESSOR OF MATHEMATICS IN THE UNITED STATES NAVY 
AND 

WILLIAM WOOLSEY JOHNSON 

PROFESSOR OF MATHEMATICS IN SAINT JOHN'S COLLEGE ANNAPOLIS MARYLAND 






REVISED EDITION. 



8' 



NEW YORK 

JOHN WILEY AND SONS 

15 Astor Place 

1879 



a *2$ 






Copyright, 1877, 
John Wiley and Sons. 



New York : J. J. Little & Co., Printers, 
10 to 20 Astor Place. 



PREFACE. 



The difficulties usually encountered on beginning the study of the 
Differential Calculus, when the fundamental idea employed is that of 
infinitesimals or that of limits, together with the objectionable use of in- 
finite series involved in Lagrange's method of derived functions, have 
induced several writers on this subject to return to the employment of 
Newton's conception of rates or fluxions. The readiness with which 
this conception is grasped, and the precision it gives to the preliminary 
definitions, promise an advantage which, however, is in most cases 
sacrificed by resorting to the use of limits in deducing the formulas for 
differentiation. 

These considerations induced the authors of this work to seek to 
derive the differentials of the functions by an analytical method 
founded upon the notion of rates, and entirely independent of the dif- 
ficult conceptions of infinitesimal increments and of limiting ratios. 

The investigation thus initiated resulted in a -satisfactory method 
of obtaining the differentials of the simple functions, which was em- 
bodied in a paper communicated to the American Academy of Arts and 
Sciences, January 14, 1873, by Professor J. M. Peirce, and published 
in the Proceedings of the Society. This paper was subsequently re- 
written and published as a pamphlet.* 

A complete resume of the original paper, by J. W. L. Glaisher, of 
Trinity College, Cambridge, was published in August, 1874, in the 
Messenger of Mathematics, Macmillan & Co., London ; vol. iv, page 58^ 

The method alluded to may be briefly described thus : — Denoting 

dx 
an assumed finite interval of time by dt, dx is so defined that — is the 

dt 

* On a new Method of obtaining the Differentials of Functions, with especial 
reference to the Newtonian conception of rates or velocities, by J. Minot Rice and W. 
Woolsey Johnson. D. Van Nostrand, Publisher, New York, 1875. 

iii 



IV PREFACE. 



expression for the rate of x (page 12) ; and, after establishing a few- 
elementary propositions, which are immediate consequences of the 

definitions, it is shown (page 17) that, whenj ; is a function of x, -~ has 

a value independent of the assumed finite value of dx. For the 
mode in which the differentials may be directly deduced, with the aid 
of this important proposition, the reader is referred to page 23 and to 
page 39. 

It is not the intention of the authors to disparage the use of the 
limit as an instrument of mathematical research. It is only claimed 
that the difficulties attending the employment of this notion are so 
great as to render it desirable to avoid introducing it into the funda- 
mental definitions of a subject necessarily involving many other con- 
ceptions new to the student. • 

The distinction between the view of the differential calculus here 
presented, and that found in most of the standard works on the sub- 
ject hitherto published, may be stated thus : — The derivative -j- is 

usually defined as the limit which the ratio of the finite quantities Ay 

and Ax approaches when these quantities are indefinitely diminished: 

when this definition is employed, it is necessary, before proceeding to 

kinematical applications, to prove that this limit is the measure of the 

relative rates of x and y. In this work the order is reversed ; that is, 

dx and dy are so defined that their ratio is equal to the ratio of the 

relative rates of x and y, and in Chapter XI, by applying the usual 

method of evaluating indeterminate forms, it is shown that the limit of 

Av . dy 

-p , when Ax is diminished indefinitely, is equal to the ratio -7-. Thus 

the employment of limits is put off until we are prepared to show that 
the limit has a definite value capable of expression in a language 
already familiar to the student. 

The early introduction of elementary examples of a kinematical 
character (see pp. 28, 37, and 57 ; also, Section X, p. 76) which this 
mode of presenting the subject permits, will be found to serve an 
important purpose in illustrating the nature and use of the symbols 
employed. 

The method of treating limits employed in Section XXXVIII is a 



pre fa a:. 



modification of the method given in M. J. Bertrand's Trait J de Calcul 
differcntitl et integral, pages 41 and 136, edition of 1864. To this 
valuable work we are also indebted for many other suggestions. 

The determination of the maxima and minima values of functions, 
and the evaluation of indeterminate forms, are so treated as to be in- 
dependent of Taylor's theorem. In the investigation of these subjects, 
and in many other cases, we have found it desirable to substitute for 
the demonstrations commonly given others more in harmony with the 
conception of rates. 

It has been found necessary to devote an unusually large amount 
of space (from page 230 to page 415) to the geometrical applications 
of the differential calculus, in consequence of the lack of available 
text-books on Curve Tracing and on Higher Plane Curves. These 
pages include Chapter IX, which consists of a brief discussion of the 
equations and many of the properties of the best-known higher plane 
curves, and is introduced chiefly for convenience of reference. We 
trust this chapter will be found a very useful feature of the book. 

To facilitate the use of this work as a text-book, it has been di- 
vided into short sections, each of which is followed by a copious col- 
lection of examples. In the arrangement of these examples the order 
of subjects in the section has been generally followed, and easy ex- 
amples usually precede the more difficult ones. 

It will not in general be found advisable for the beginner to solve 
all the examples on reading this work for the first time. They occupy 
nearly one third of the entire volume, and are intended as a collection 
from which the instructor may select at his discretion. 

Many of these examples were prepared especially for this work. 
The others were taken chiefly from the collections of G?-egory 
(Walton's edition), Frenet, and Tisserand j and from the treatises of 
Todhu?iter, Williamson, and Connell. 

A preliminary edition of this work in three parts has been in use for 
several years at the U. S. Naval Academy and elsewhere. Part I was 
printed by Messrs. John Wiley and Son, New York, 1874 ; Part II 
and Part III were printed at the Government Printing Office, Wash- 
ington, D. C, in 1875 and 1876. To form the volume now issued, 
these three parts have been rewritten, and much new matter has 
been added. 



VI PREFACE. 



A Limited Course. 

When the time allotted to the Differential Calculus is insufficient for 
a more extended course, the following articles may be taken. The 
text of these articles occupies only about 140 pages. This course may 
be supplemented by selections from Chapter IX. Appropriate exam- 
ples will be found at the beginning of the collections which accompany 
each section. 

Art. 1-81, 93-101, 107-112, 119-139, 147-155, 159, 161-167, 200- 
209, 241-245, 311-326, 33 2 -34i, 353-357, 3 8 9-394> 39 8 -4<H, inclu- 
sive. 

J. M. R. 

W. W. J. 

Annapolis, Maryland, September, 1879. 



CONTENTS. 



CHAPTER I. 

Functions, Rates, and Derivatives. 

I. 

PAGE 

Functions I 

Implicit functions 3 

Inverse functions 4 

Classification of functions 4 

Expressions involving an unknown function 5 

Examples 1 6 

II. 

Rates g 

Constant rates 10 

Variable velocities 1 1 

Illustration by means of Attwood's machine n 

The measure of a variable rate 12 

Differentials 12 

The differentials of polynomials 13 

The differential of mx 14 

Examples II 15 

III. 

The differentials of functions 16 

The derivative — its value independent of dx 17 

The geometrical meaning of the derivative 19 

Examples III 21 

CHAPTER II. 

The Differentiation of Algebraic Functions. 

IV. 

The square 23 

The square root 25 

Examples IV 26 

vii 



vill CONTENTS. 



V. PAGE 

The product 29 

The reciprocal 30 

The quotient 31 

The power 32 

Examples V ; 34 

CHAPTER III. 
The Differentiation of Transcendental Functions. 

VI. 

The logarithm 39 

The Napierian base 41 

The logarithmic curve {y = log e .r) 42 

Logarithmic differentiation 43 

Differentials of algebraic functions deduced by logarithmic differentiation 44 

Exponential functions 45 

Examples VI 46 

VII. 

Trigonometric or circular functions 50 

The sine and the cosine 51 

The tangent and the cotangent 52 

The secant and the cosecant 53 

The versed sine 53 

Examples VII > 54 

VIII. 

Inverse circular functions — their primaiy values. 58 

The inverse sine and the inverse cosine 60 

The inverse tangent and the inverse cotangent 61 

The inverse secant and the inverse cosecant 62 

The inverse versed-sine °3 

Examples involving trigonometric reductions 63 

Examples VIII 64 

IX. 

Differentials of functions of two variables 67 

Derivatives of implicit functions 69 

Examples IX 7 1 

Miscellaneous examples of differentiation 7 2 



CONTENTS. IX 



CHAPTER IV. 
Successive Differentiation. 

X. PAGE 

Velocity and acceleration 76 

Component velocities and accelerations 78 

Examples X 79 

XL 

Successive derivatives 82 

The geometrical meaning of the second derivative 82 

Points of inflexion 83 

Successive differentials 84 

Equicrescent variables 84 

Implicit functions 85 

Differential equations 86 

Examples XI 88 

XII. 

Expressions for the wth derivative . 92 

Leibnitz' theorem 95 

Numerical values of successive derivatives 98 

Examples XII 99 

CHAPTER V. 
The Evaluation of Indeterminate Forms. 

XIII. 

Indeterminate or illusory forms 104 

Evaluation by differentiation 105 

Examples involving decomposition 107 

Examples XIII 109 

XIV. 

The form 5° 113 

Derivatives of functions which assume an infinite value 115 

Algebraic fractions, x being infinite 117 

The form o.co 117 

The form 00 — 00 118 

Examples XIV 119 



CONTENTS. 



XV. PAGE 

Functions whose logarithms take the form o- oo 122 

The form i 00 122 

The form o° 123 

Functions having two limiting values for a single value of x 124 

Examples XV '. . . 126 

XVI. 

Indeterminate forms of functions of two variables 129 

Evaluation of derivatives of implicit functions 130 

Evaluation of derivatives when x = o and y = o 131 

Examples XVI 132 

CHAPTER VI. 

Maxima and Minima of Functions of a Single Variable. 

XVII. 

Conditions indicating the existence of maxima and minima 136 

Maxima and minima of geometrical magnitudes 138 

Examples XVII 140 

XVIII. 

Method of discriminating between maxima and minima 144 

Alternate maxima and minima 145 

The employment of a substituted function 147 

Examples XVIII 149 

XIX. 

Employment of derivatives higher than the first 151 

Complete criterion for a maximum or a minimum 153 

Examples XIX 155 

XX. 

Maxima and minima of implicit functions 157 

Infinite values of the derivative 159 

Examples XX 162 

Miscellaneous examples of maxima and minima 163 

CHAPTER VII. 
The Development of Functions' in Series. 

XXI. 

The nature of an infinite series I7 1 

Convergent and divergent series 173. 



CONTENTS. xi 



■ 



Taylor's theorem 174 

Lagrange's expression for the remainder 176 

A second expression for the remainder 178 

Limits to the application of Taylor's theorem 179 

The binomial theorem 180 

Examples XXI 182 

XXII. 

Maclaurin's theorem 185 

The exponential series and the value of £ 186 

Logarithmic series 188 

Computation of Napierian logarithms 189 

The modulus of tabular logarithms igi 

The developments of sin x and of cos x 191 

Huygens' approximation to the length of a circular arc 192 

Examples XXII 193 

XXIIL 

Even functions and odd functions % . 196 

The development of the inverse tangent 197 

Maclaurin's theorem applied to implicit functions 198 

Evaluation by development in series 199 

The formula for evaluation deduced by means of Taylor's theorem 201 

Symbolic form of Taylor's theorem 201 

Examples XXIII 202 

XXIV. 

Development of functions by means of differential equations 204 

The computation of tc 206 

Examples XXIV 209 

XXV. 

Functions of imaginary quantities 212 

Logarithms of imaginary and of negative quantities 213 

Hyperbolic functions 214 

De Moivre's theorem 215 

Multiple values of zzth roots 216 

Resolution of certain expressions into factors 217 

The sine and the cosine expressed as continued products 219 

Bernoulli's numbers 221 

The development of B cot 224 

Bernoulli's numbers expressed by means of numerical series 225 

Examples XXV 226 



xn CONTENTS. 



CHAPTER VIII. 

Curve Tracing. 
XXVI. 



PAGE 



Equations in the form y = fix) 230 

Asymptotes parallel to the coordinate axes 230 

Minimum ordinates and points of inflexion 232 

Oblique asymptotes 233 

Curvilinear asymptotes 235 

Examples XXVI 236 

XXVII. 

Equations in the form fix, y) = o 239 

The employment of an auxiliary variable 240 

Maxima and minima of the coordinates in terms of an auxiliary variable 243 

Tangents and points of inflexion 244 

Nodes and cusps at the origin 246 

Isolated points satisfying the conditions which usually determine a cusp 248 

Multiple points 248 

Equivalent number of nodes 249 

Examples XXVII 249 

XXVIII. 

Points at infinity 251 

Asymptotes 253 

The points at which an asymptote cuts the curve 255 

Asymptotes parallel to the coordinate axes 256 

Parabolic branches 2 57 

Parallel asymptotes 2 59 

Nodes at infinity 2 ^ 2 

The characteristic of Newton's trident 2 ^4 

Asymptotes to imaginary branches 2 "4 

Examples XXVIII 26 5 

XXIX. 

Curves given by polar equations 2 ^8 

Asymptotes determined by means of polar equations 270 

Asymptotic circles. 2 7 J 

Examples XXIX 2 7 2 



CONTENTS. 



XXX. PAGE 

Transcendental curves 275 

Stop points 275 

Salient points 276 

Branches pointillees 277 

Examples XXX 278 

CHAPTER IX. 
Equations and Constructions of Certain Higher Plane Curves. 

XXXI. 

The parabola of the «th degree 281 

The cubical and the semicubical parabolas 282 

The cissoid of Diodes 283 

The duplication of the cube by means of the cissoid 284 

The conchoid of Nicomedes 285 

The trisection of an angle by means of the conchoid . 286 

The quadratrix of Dinostratus 286 

The witch of Agnesi 288 

The folium of Descartes 289 

The strophoid or logocyclic curve 290 

The limacon of Pascal 292 

The trisectrix. 293 

The cardioid 294 

The Cartesian ovals 295 

The Cassinian ovals 299 

The lemniscata of Bernoulli « 301 

The spiral of Archimedes 302 

The hyperbolic or reciprocal spiral 302 

The lituus 303 

The logarithmic or equiangular spiral 304 

The loxodromic curve and its projections 304 

The parabolic spiral 307 

The logarithmic or exponential curve 308 

The sinusoid 308 

The cycloid 309 

The companion to the cycloid 311 

The trochoid 311 

The prolate and the curtate cycloids 312 

The epicycloid 313 

The two-cusped epicycloid 314 

The epitrochoid 315 



xiv CONTENTS. 



PAGE 

The limacon and the cardioid cases of the epitrochoid 316 

The hypocycloid and the hypotrochoid 317 

The ellipse a case of the hypotrochoid 318 

The limacon and the cardioid cases of the hypocycloid 318 

The four-cusped hypocycloid 320 

The involute of the circle 321 

The catenary 322 

The tractrix 3 2 3 

Curves of pursuit 323 

Roulettes 324 

Inverse curves 3 2 4 

Pedal curves 326 

Reciprocal curves 3 2 7 

The inverse and pedal of the conic 328 



CHAPTER X. 

Applications of the Differential Calculus to Plane Curves. 

XXXII. 

The equation of the tangent 3 2 9 

The equation of the normal , 33 1 

The equation of the tangent to a conic 33 1 

Subtangents and subnormals 33 2 

The perpendicular from the origin upon a tangent 334 

Examples XXXII 335 

XXXIII. 

Polar coordinates ' • • • 337 

Polar subtangents and subnormals 339 

The perpendicular from the pole upon a tangent 34° 

The perpendicular upon an asymptote ~ 34 1 

Points of inflexion 34 1 

The polar equation of a pedal 343 

Discussion of the curve r n = a" 1 cos mO 345 

Examples XXXIII 347 

XXXIV. 

Curvature 35 l 

The direction of the radius of curvature 353 

The radius of curvature in rectangular coordinates 354 

The radius of curvature of a meridian 355 



CONTENTS. xv 



PAGE 

The radius of curvature at the origin 355 

Expressions for p in which x is not the independent variable 358 

The radius of curvature in polar coordinates 361 

Relations between p, p, and r 363 

The chord of curvature through the origin 365 

Examples XXXIV 366 

XXXV. 

Evolutes 573 

The length of an arc of the evolute 378 

Involutes and parallel curves 378 

The radius of curvature at a cusp and at a point of inflexion 379 

The radius of curvature of the evolute 380 

Evolutes determined by means of the relation between p and </> 381 

Examples XXXV 382 

XXXVI. 

Envelopes 385 

Fixed points of intersection in a system of curves 389 

Envelopes determined by means of equations in an irrational form 391 

Two variable parameters 393 

The application of undetermined multipliers 395, 

Examples XXXVI 397 

XXXVII. 

Envelopes of systems of straight lines 400 

The evolute regarded as an envelope 402 

Negative pedals 403 

Reciprocal curves 405 

The reciprocal of a conic 406 

Caustics. . 408 

Examples XXXVII 409 

CHAPTER XI. 
Functions of Two or More Variables. 

XXXVIII. 

The derivative regarded as the limit of a ratio 415 

Partial derivatives 416 

The approximate values of errors due to small errors of observation 419 

Examples XXXVIII 422 



XVI CONTENTS. 



XXXIX. 

PAGE 

The second derivative regarded as a limit 424 

Higher partial derivatives 426 

Commutative character of independent differentiations 427 

Symbols of operation 429 

Commutative and distributive operations 430 

Symbolic transformations 432 

Euler's theorems concerning homogeneous functions 434 

Examples XXXIX 435 

XL. 

Change of the independent variable 439 

Transformations of certain operative symbols 441 

Expressions involving partial derivatives. 442 

_ . . . d'hi d' 2 u 

1 ransformation of -— r H — r-^ 445 

dx- dy~ 

_, . d'-u d' 2 u d^u 

T. ransformation of —=-z H — rr ^ — rr 44o 

dx- dy~ dz~ 

Examples XL 448 

XLI. 

Lagrange's theorem 454 

Laplace's theorem . . .' 458 

Development of functions of two variables in series 459 

Symbolic expression for the series 461 

Maxima and minima of functions of two variables 461 

Lagrange's condition 463 

Exa?nples XLI , 466 



THE 

DIFFERENTIAL CALCULUS 



CHAPTER I. 

Functions, Rates, and Derivatives. 



I. 

Fzcnctions. 

1. A QUANTITY which depends for its value upon another 
quantity is said to be a function of the latter quantity. Thus 
x 2 , tan x, log {a + x), and a x are functions of x. 

The quantity upon which the function depends must be 
regarded as variable, and be represented in the analytical 
expression for the function by an algebraic symbol. This 
quantity is called the independent variable. It is essential 
that variation of the independent variable should actually 
produce variation of the function. Thus the quantities 
x°, x 2 + {a + x) {a — x), and (tan x + cot x) sin 2x are not func- 
tions of „r, since each admits of expression in a form which 
does not involve x. 

2. The notation f(x) is employed to denote any function 
of x, and, when several functions of x occur in the same in- 



2 FUNCTIONS RATES AND DERIVATIVES. [Art. 2. 

vestigation, such expressions as Fix), F' (x), (x), etc., are 
employed, the enclosed letter always denoting the indepen- 
dent variable. When expressions like f(i),f(a\ f(2x), or 
fip) are employed, it must be understood that the enclosed 
quantity is to be substituted for x in the expression which 
defines fix). Thus, if we have 

fix) = X* + X, 

/(i) = 2, f(2x) = 4^r 2 + 2x y and /(o) = o. 
Again, if Fix) = log a x, 

F(i) = o, .F(o) = — oo, and F(a) = I. 

3. When x denotes the independent variable upon which 
a function depends, any quantity independent of x is, in con- 
tradistinction, called a constant ; both when it is an absolute 
constant, like I, \/2, or 7r, and when it is denoted by a symbol, 
like a, u, or y, to which any value, can be assigned. Thus, 
when a* is denoted by f{x), it is considered simply as a func- 
tion of x, and a is regarded as a constant. 

When it is desired to express that a quantity is a function 
of two quantities, both the symbols denoting them are placed 
between marks of parenthesis. Thus, since a* is a function of 
x and a, we may write 

fix, a) — a*. 
Accordingly we have 

/O, S) = P, AS, 2) = 8, and /(2, 3) = 9- 

4. It is often convenient to represent the value of a func- 
tion of x by a single letter ; thus, for example, y = jit 2 . When 
this notation is used, if we represent the independent variable 
x by the abscissa of a point, and the function y by the corre- 



§ I.] IMPLICIT FUNCTIONS. 



sponding ordinate, a curve may be constructed which will 
graphically represent the function, and will serve to illustrate 
its peculiarities. 

Rectangular coordinates are usually employed for this 
purpose. See diagram, Art. 10. 

A function of the form 

y = m x + b, 

m and b being constants, is represented by a straight line. 
Functions of this form are, for this reason, called linear func- 
tions. 

Implicit Functions. 

5« When an equation is given involving two variables x 
and y, either variable is obviously a function of the other ; 
and the former variable, when its value is not directly ex- 
pressed in terms of the other, is said to be an implicit func- 
tion of the latter. Thus, if we have 

ax* — $cixy + y* — a 3 = o, 

either variable is an implicit function of the other. 
By solving the above equation for x, we obtain 

2 r \ 4 a J 

In this form of the equation, x is said to be an explicit func- 
tion of y. 

This example will serve to illustrate the fact, that from a 
single equation involving two variables, there may be derived 
two or more explicit functions of the same variable. In the 
above case, x is said to be a two-valued function of y; while, 
since the equation is of the third degree in y, the latter is a 
three-valued function of x. 



FUNCTIONS RATES AND DERIVATIVES. [Art. 6. 



Inverse Functions. 

6. If y = fix), x is some function of y ; we may therefore 
write 

y = fix), whence x = </>(jk). 

Each of the functions f and </> is then said to be the inverse 
function of the other. Thus, if 

y = a*, we have x = log a y ; 

hence each of these functions is the inverse of the other. So 
also the square and the square root are inverse functions. 

7. In the case of the trigonometric functions, a peculiar 
notation for the inverse functions has been adopted. Thus, 
if we have 

x = sin 0, we write 6 = sin _1 ^r. 

Whenever trigonometric functions are employed in the 
Calculus, the symbol representing the angle always denotes 
the circular measure of the angle ; that is, the ratio of the arc 
to the radius. Hence sin-\r maybe read either "the in- 
verse sine of x" or " the arc whose sine is x." 

The inverse trigonometric functions are evidently many- 
valued. See Art. 54. 

The Classification of Functions. 

8. With reference to its form, an explicit function is 
either algebraic or transcendental. 

An algebraic function is expressed by a definite combination 
of algebraic symbols, in which the exponents do not involve 
the independent variable. 



§ I.] THE CLASSIFICATION OF FUNCTIONS. 5 

All functions not algebraic are classed as transcendental. 
Under this head are included exponential functions ; that is, 
those in which one or more exponents are functions of the 
variable, as, for example, a?, xa Vx t etc. : logarithmic func- 
tions : the direct and inverse trigonometric functions, and 
other forms which arise in the higher branches of mathematics. 

9. With reference to its mode of variation, a function is 
said to be an increasing function when it increases and de- 
creases with x ; and a decreasing function when it decreases 
as x increases, and increases as x decreases. Thus, it is evi- 
dent that x 3 is always an increasing function of x, while - is 

always a decreasing function of x%. Again, tan.r is always an 
increasing function, but sin,r is sometimes an increasing and 
sometimes a decreasing function of x. 

10. The increase and decrease here considered are aire- 
braic. For example, x* is an increasing function when x is 
positive, but when x is negative it becomes 
a decreasing function ; for, when x is negative 
and algebraically increasing, x~ is decreasing. 

The curve y = x~ which illustrates this 
function is constructed in Fig. i. Since alge- 
braic increase in the value of x is represented 
by motion from left to right, whether the 
moving point is on the left or on the right of 
the axis of y, the downward slope of the curve on the left 
of the origin indicates that x~ is a decreasing function when x 
is negative. 

Expressions involving an Unknown Function. 

II- An expression involving f(x), as, for example, xf(x) 
or F[f(x)], is generally a function of x\ but it may happen 




6 FUNCTIONS RATES AND DERIVATIVES. [Art. II. 

that such an expression has a value independent of x. Thus, 
suppose that, in the course of an investigation, the following 
equation presents itself : — 

xf{x) = zf(z\ 

in which / denotes an unknown function, and x and z are en- 
tirely independent arbitrary quantities. When this is the case, 
we can make z a fixed quantity, and give to x any value what- 
ever; that is, we can make x a variable and z a constant; 
but if z is a constant, zf{z) is likewise a constant, we can, 
therefore, write 

xf{x) = c, 



c being an unknown constant. Hence Ave have 



x 



The value of the constant c is readily found, if we know the 
value of f{x) corresponding to any one value of x. 

Examples I. 

I. (a) For what value of n does x n cease to be a function of x? 
(/3) For what values of x does it cease to be a function of n? 

(a) When n = o. (,3) When x = I, or x — o. 

2 If y ( i _ _- ~ x \ = x + a ' X ~ "* -, show that y is a function of a, but 
\ • a + x] a + x 

not of x. 

3. Show that sin,r tan \x + cos.r is not a function of x. 

4. If y == x + \/{i + x 2 ), show that/ 2 — 2xy is not a function of x. 

5. If f(x) = x\ find the value of f{x + k) ; of f{2x); of fix' 1 ) ; of 
/•(**-*)■; of/(i);/(i2);/[/(x)]. 

/(^ + ^) = x 2 + 2/z ^r + /^ 2 . 



§ l.j EXAMPLES OF FUNCTIONS. 7 

6. If/(0) = cos 0, find the value of/(o); of /(£*■); of /(£*); of 

7. If F{x) =a*, give the value of F(a)\ of F(i) ; of F(p). Also 
show that in this case [F(x)f = F(2x), 

8. Given/ 2 — 2#_y + .r 2 = o, make y an explicit function of x. 

y = a ± ^{ar — xj. 

9. Given 1 4- \og a y = 2 log a (x 4- a), make y an explicit function of x. 

(x + af 
y= 

10. Given the equations — 

n + 1 =n (cos 2 0' 4- cos#' cos 6 4- cos 2 0), 
and ii — \—n (sin 2 0' + sin 6' sin 4- sin 2 0) ; 

eliminate n, and make 6' an explicit function of Q. Also make 11 an ex- 
plicit function of 6. 



)' = 5±|ir, and 11 = + 



sin cos0 



11. Given sin - x x+ sin" : ji/ = «, make y an explicit function of x. 

y = sin or 4/(1 — -r 2 ) — .r cos a. 

12. Given tan - l x + tan - *j/ = a, make^y an explicit function of x. 

tan a — x 
y ~ 1 +x tana' 

13. Given xy — 2.r + y = 11, show that y is not a function of x when 
n — 7.. 

2,x I 

14. If y = , show that the inverse function is of the same 

form. 

1 4- x 

15. If y =/(>) = — — --, find 2 =/(y), and express ^ as a function 

of x. 1 



16. If both f and ^ denote increasing functions, or, if both denote 
decreasing functions, show that <?[f{x)} is an increasing function. 
Also show that the inverse of an increasing function is an increasing 
function. 



8 FUNCTIONS RATES AND DERIVATIVES. [Ex. I. 

17. Find the inverse of the function, y = log £ \x + 4/(1 + X s )]. 

x=±(e>-e-v). 

18. II f(x) be an unknown function having the property 

prove that f{\) = o. 

Futy = 1. 

19. If f(x) has the property 

f{x+y)=f{x)+f{y), 

prove that/(o) = o. Also prove that the function has the property 

f{px)=pf(x), 

in which p is a positive or negative integer. 

For positive integers, put y = x, 2x, $x, etc., in the given equation ; for 
negative integers, put y = — x. 

20. If /denotes the same function as in Example 19, prove that 

f(mx) = mf(x), 
m denoting any fraction. 

Solution : — 

P 
Putting z = —x, qz=px, 

f(?z)=f(P*); 

hence, by Example 19, qf{z) = Pf (•*)» 

or /(*)?= £/.(*), 



J \q ! q J 



21. Given, the property of the same function proved in Example 20; 

viz., 

f{mx) = mf{x) ; 



§ I.] EXAMPLES OF FUNCTIONS. 

by putting- z for m x, show that 

i/CO = j/W. 

and thence deduce the form of the function. See Art. it. 

f(x) = ex. 

22. Given, [9 (*)]■* = [9 (s-)]* , and 9(1) = e, 
determine 9 (.r). 

23. Given $ (.r) + $ (y) = $ (xy) 
prove 9 {x»<) = ;/z 9 (>), 
and thence prove $ (.r) = c log.r. 

£/><? //^ methods of Examples 19, 20, and 21. 



f(x) = e* 



II. 

12. In the Differential Calculus, variable quantities are 
regarded as undergoing continuous variation in magnitude, 
and the rates of variation, denoted by appropriate symbols, 
are employed in connection with the values of the variables 
themselves. 

If a varying quantity be represented by the distance of a 
point moving in a straight line from a fixed origin taken on 
that line, the velocity of the moving point will represent the 
rate of increase or decrease of the varying quantity. 



Fig. 2. 

Thus O (Fig. 2) being the fixed origin and OP a variable 
denoted by x, P is the moving point whose velocity repre- 
sents the rate of x. The velocity of P, or the rate of x, is 
regarded as positive when P moves in the direction in which 
x increases algebraically ; thus, taking the direction OX, or 
toward the right, as the positive direction in laying off x, the 



IO . FUNCTIONS RATES AND DERIVATIVES. [Art. 12. 

velocity is positive when P moves toward the right, whether 
its position be on the right or on the left of the origin. Ac- 
cordingly, a rate of algebraic decrease is considered as nega- 
tive, and would be represented by a point moving toward the 
left. 

Constant Rates. 

13. The rate of a quantity like the velocity of a point may 
be either constant or variable. A velocity is uniform or con- 
stant, when the spaces passed over in any equal intervals of 
time are equal, or, in other words, when tlie spaces passed over 
in any intervals of time are proportional to the intervals. 

The numerical measure of a uniform velocity is the space 
passed over in a unit of time ; then if / denote the time elapsed 
from an assumed origin of time, and k the space passed over 
by a moving point in a unit of time, kt will denote the space 
passed over in the time t. Hence, whenever the velocity is 
uniform, the quotient obtained by dividing the number of 
units of space by the number of units of time occupied in 
describing this space is constant, and serves as the numerical 
measure of the velocity. 

14- Now, if x be a quantity having a uniform rate k, it 
will be represented by the distance from the origin of a point 
having the uniform velocity k, and if a denote the value of x 
when t is zero, we shall have 

x = a + kt. . . . . . . . (i) 

This formula expresses a uniformly varying quantity as a 
function of t. When x is a uniformly decreasing quantity, 
k is, of course, negative. 

Conversely, if x, when expressed as a function of t, is of the 
form (i), involving the first power only of /, then x is a quan- 
tity having a uniform rate, and the coefficient k is a measure 
of this rate. 



II.] VARIABLE VELOCITIES. II 



Variable I 'c loci tics. 

15. If the velocity of a point be not uniform, its numerical 
measure at any instant is the number of units of space which 
would be described in a unit of time, were the velocity to remain 
constant from and after t lie given instant. 

Thus, when we speak of a body as having- at a given in- 
stant a velocity of 32 feet per second, we mean that should the 
boclv continue to move during the whole of the next second, 
with the same velocity which it had at the given instant, 32 
feet would be described. The actual space described may be 
greater or less, in consequence of the change in velocity which 
takes place during the second ; it is, for instance, greater than 
the measure of the velocity at the beginning of the second, 
in the case of a falling body, because the velocity increases 
throughout the second. 

16. Att wood's machine for determining experimentally 
the velocities acquired by falling bodies furnishes a familiar 
example of the practical application of the principle em- 
bodied in the above definition. 

This apparatus consists essentially of a thread passing 
over a fixed pulley, and sustaining equal weights at each ex- 
tremity, the pulley being so constructed as to offer but slight 
resistance to turning. On one of the weights a small bar of 
metal is placed, which, destroying the equilibrium, causes the 
weight to descend with an increasing velocity. To deter- 
mine the value of this velocity at any point, a ring is so placed 
as to intercept the bar at that point, and allow the weight to 
pass. Thus, the sole cause of the variation of the velocity 
having been removed, the weight moves on uniformly with 
the required velocity, and the space described during the 
next second becomes the measure of this velocity. 



1 2 FUNCTIONS RA TES AND DERI VA TIVES. [Art. I J. 

Variable Rates. 

17. When x is a function of t, but not of the form ex- 
pressed by equation (i), Art. 14— that is, when the function is 
not linear— the rate of x will be variable. To obtain the 
measure of this rate at any given instant, we employ the 
same principle as in the case of a variable velocity. Thus, 
let x be represented by OP as in Fig. 2, Art. 12, let the sym- 
bol dt denote an assumed interval of time, and let dx denote 
the space which would be described in the time dt, were P 
to move with the velocity which it has at the given instant 
unchanged throughout the interval of time dt. Then the 
space which would be described in a unit of time is, evidently, 

dx 
It 1 

which is therefore the measure of the velocity of P, or the 
rate of x. 

This ratio is in general variable, but, when x is of the form 
a 4- kt, it has been shown in Art. 14 that k is the measure of 
the rate ; we therefore have 

dx . . 

—r- = k. when x = a + kt. 

at 

Differentials. 

18. The quantities dx and dt are called respectively "the 
differential of x" and "the differential of t." 

In accordance with the definition of dx given in the pre- 
ceding article, the differential of a variable quantity at any 
instant is the increment which would be received in the time 
dt, were the quantity to continue to increase uniformly 
during that interval of time with the rate it has at the given 






§ II.] DIFFERENTIALS. 13 

instant. The quotient obtained by dividing the differential of any 
quantity by dt is therefore the measure of the rate of the quantity. 
The differential of a quantity is denoted by prefixing d to 
the symbol denoting the quantity ; when the symbol denot- 
ing the quantity is not a single letter it is usually enclosed 
by marks of parenthesis to avoid ambiguity. Thus, d(x'), 
d(xy), ditimx), d{a* + x n ), etc. 

The Differentials of Polynomials. 

19. Let x and y denote two variable quantities, and let a 
and b denote particular simultaneous values of x and y, while 
k and k' denote corresponding values of the rates of x and y. 

Now, if x and y should continue to vary with these rates, 
their values would (see Art. 14) be expressed by 





x = a + k t, 


and 


y —b-r k't, 


whence 


x+y = a + b+{k+ k') t. 



Thus the quantity x + y would become a uniformly varying 
quantity, and, by Art. 14, its rate would be k + k', which, 
therefore, is the measure of the rate of x +y at the instant 
when x and y have the rates k and k' . Consequently, 



dt dt ' dt 

Now, since k and k' denote any values of the rates, this equa- 
tion is universally true. We have, therefore, 

d(x+y) = dx + dy (1) 

This formula is easily extended to the sum of any number 
of variables. Thus, 

d(x+y + js + - • •) = dx + d(y+s+* • •) =zdx + dy + dz+ (2) 



14 FUNCTIONS RATES AND DERIVATIVES. [Art. 20. 

20. The differential of a constant is evidently zero, hence 

d(x+h) = dx (3) 

Again, if y = — ar, 7 + ^ = 0, 

hence, by equation (1), since zero is a constant, we have 

dy 4- dx = o, or ^/ = — <£r ; 

that is, d(—x) = — <£r (4) 

The differential of a negative term is therefore the negative 
of the differential of the term taken positively. 

It appears, on combining the results expressed in equations 

(2), (3), and (4), that the differential of a polynomial is the alge- 
braic sum of the differentials of its terms ; and that constant 
terms disappear from the result. 

The Differential of a Term having a Constant 
Coefficient. 

21. Let the term be denoted by mx, m denoting a con- 
stant. 

Resuming equation (2), Art. 19 ; viz., 

d{x+y + z+ • • •) — dx + dy+dz+ • • •, 
and denoting the number of terms by /, we put 

x—y == z = , 

thus obtaining d(px) —pdx, . (1) ' 

p denoting an integer. 



§ II.] THE DIFFERENTIAL OF (m x). 1 5 

To extend equation (i) to the case in which m denotes a 
fraction, let 

z = - x, then qz = px. 

q 

By applying equation (i) we obtain 

q dz —pdx, or dz — — dx ; 



2 



that is, dl-x) = -^r. 



Hence generally, when m is positive, 

d{inx)-=mdx (2) 

Since d{— x) — — dx, this equation is true likewise when m 
is negative. 

It therefore follows that the differential of a term having a 
constant coefficient is equal to the product of the differential of the 
variable factor by the constant coefficient. 



Examples II. 

1. Find the differential of — , and of . , _ 

■\a m — 2 2dx , ax 

, and 

3<z m — 2 



1 , ,.rr ■ , . x — a a — x 

2. Find the differential of — , and of 



in 2 dx dx 

— 2, and a- 

m- vv 



„ , n . ~ ., , a + b + (a — b)x dx 

3. Find the differential of s -^ • -—7 - 

J a 2 — b 2 a + b 

4. Find the differential of —"*» and of —7 — —7- ■ 

a + b a(a + b) 

dx , and b ^ dx + d y) , 



1 6 FUNCTIONS RATES AND DERIVATIVES. [Ex. II. 

dy 

5. Given ay + bx + zcx + a£ = o, to find —7-. 7 

J dx dy b + 2.c 

dx ~ ~ # 

6. Given j/ log a + x sin a: — j cos a — ax + tan a = o, to find —* 

dy a — sin a 



dx log a — cos a 

7. Given ay cos 2 a: — 2^ (1 — sina).r = b(a — x cos 2 a), to find^. 

dx 

dy_b (1 — sin a) 
dx ~ a (1 + sin a)' 

8. Given a 1 + 2 (1 + cos a) j = (x + j) sin 2 or, to find -~. 

dy 2 a 

~r = tan 2 — • 
dx 2 

x y z 

9. Given — I- j + — = 1, to express dz in terms of dx and <^>/. 

dz = dx — — dy. 

a b J 

10. A man whose height is 6 feet walks directly away from a lamp- 
post at the rate of 3 miles an hour. At what rate is the extremity 
of his shadow travelling, supposing the light to be 10 feet above the 
level pavement on which he is walking? 

Draw a figure, and denote the variable distance of the man from the 
Ia?np-post by x, and the distance of the extremity of his shadow from the 
post by y. 7i miles per hour. 

11. At what rate does the man's shadow (Ex. 10) increase in length ? 



III. 

Differentials of Functions of an Independent Variable. 

22. When the variables involved in any mathematical 

investigation are functions of an independent variable x, the 

dx 
latter may be assumed to have a rate denoted by — -, in which 

at 



§ III.] THE DERIVATIVE. 1/ 

dx is arbitrary. So also the corresponding rate of y will be 
denoted bv — , and, if y is a function of x, the value of dy will 

" dt 

depend in part upon the assumed value of dx. 

To differentiate a function of x is to express its differential 
in terms of x and dx. 

It is to be understood, of course, that the differentials 
involved in an equation are all taken with reference to the 
same value of dt. 

If two quantities are always equal, their simultaneous 
rates are evidently equal; and hence their differentials are 
likewise equal. We can therefore differentiate an equation ; 
that is, express the equality of the differentials of its mem- 
bers; provided the equation is true for all values of the 
variables involved. Thus, from the identical equation 

(x + // ) 2 = x 2 + 2/1 x + h\ 

it follows that d[(x + h) 2 ] = d(x 2 ) + 2/1 dx. 

The Derivative. 

23. Before proceeding to the differentiation of the vari- 
ous functions of x, it is necessary to show that, it 

*=/(*), a) 

the ratio -j- 

dx 

has a definite value for each value of x, independent of the assumed 
value of dx. 

Let a particular value of x be denoted by a, and let the 
corresponding value of dx be an arbitrary quantity. 

Now, although dx is arbitrary, since dt is likewise 
arbitrary, the rate of x, that is, the ratio 

dx , x 

w (2) 



1 8 FUNCTIONS RATES AND DERIVATIVES. [Art. 23. 

may be assumed to have a certain fixed value at the instant 
when x = a. The corresponding value of the rate of y, 
denoted by 

I a> 

evidently depends solely upon the rate of x and upon the form 
of the function /in equation (1). Hence, when the value of 
the rate (2) is fixed, the value of (3) is also definitely fixed. 
Denoting these fixed values by k and k\ we have, when 

x = a, 

dx , , dy ,, . dy k' 

—-= k, and -f = k , whence -f- = - • 
a/ at dx k 

Hence, corresponding to a particular value a of x, there 

exists a determinate value ->of the ratio -/-> notwithstand- 

k dx 

ing the fact that dx has an arbitrary value ; in other words, 
the value of the ratio — ^- is independent of the arbitrary value ofdx. 



24. It is obvious that, in general, this ratio will have 
different values corresponding to different values of x, and 
hence that it may be expressed as a function of x, and de- 
noted by f\x) ; thus, — 

£="-« "« 

The form of this new function / / will evidently depend upon 
that of the given function / 

The function /^.r) is called the derivative oif(x), and, since 
equation (1) may be written in the form 

dy=f'(x)dx, 

it is also called the differential coefficient of y regarded as a 
function of x. 



§ III.] GEOMETRICAL ME, IX EYC OF THE DERIVATIVE. 1 9 

When, however, the given function f(x) is of the linear 

form 

y = m x + b y 

the derivative is no longer a function of x, but is a constant, 
since the value of y gives 

dy = m dx y 

dy 

s = m. 



The Geometrical Meaning of the Derivative. 

25. Representing the corresponding values of x and y by 
the rectangular coordinates of a moving point, if this point 
move in a uniform direction, so as to describe a straight line, — 

that is, if y be a linear function of x. — the value of —- will be 

ax 

constant, by the preceding article. Hence, in the general 
case, when this ratio is variable, the point will move in a vari- 
able direction. 

If we denote the inclination of this direction to the axis of 
x by <f>, the value of </> will vary with the value of x, and the 
point will describe a curve. 

The tangent line to a curve is defined as follows : — 

The tangent to a curve at any point is the straight line which 
passes through the point , and has the direction of the curve at that 
point* 

Hence, for any point of the curve, denotes the inclina- 
tion to the axis of x of the tangent line at that point. 



* It will be shown hereafter (Art. 49) that, in the case of the circle, this 
general definition of a tangent line agrees with that usually given in Plane 
Geometry. 



20 



FUNCTIONS RA TES AND DERI V A TIVES. [Art. 26. 



26. Now, if a point, at first moving- in the curve, should, 
after passing the point whose abscissa is a, so move that the 

rates —j~ and —7- retain the values which they had at the in- 
stant of passing the given point, the direction of its motion 
will become constant, and the point will describe a straight 
line tangent to the curve at the given point. 

The value of dx may be repre- 
sented by an arbitrary increment of 
x as in Fig. 3 ; the value of dy will 
then be represented by the corre- 
sponding increment which would be 
received by y, were the point moving 
in the tangent line, as indicated in 




Fig. 3. 
the diagram. Hence 



dy 



which is evidently independent of the assumed value of dx. 
It follows that the value of the derivative of fix), for any 
value of x, is represented by the trigonometric tangent of 
the inclination to the axis of x of the curve y =f(x) y at the 
point corresponding to the given value of x. 

27- The moving point, which is conceived to describe 
the curve, may pass over it in either of two directions differ- 
ing by 180 . The two corresponding values of <f> give, how- 
ever, the same value of tan</>, since 

tan(0 ± 180 ) = tan0. 



Thus, in Fig. 3, the point P may be regarded as moving 
so as to increase x and y, in which case both dx and dy will 
be positive, and </> will be in the first quadrant ; or P may 



§ III."] GEOMETRICAL MEANING OF THE DERIVATIVE. 21 

move in the opposite direction, making dx and dy negative, 
and placing </> in the third quadrant. In either case, ~ ~- or 
tan0 is positive. 

28. It is evident that when f{x) is an increasing func- 
tion, as in Fig. 3, -j- is positive, and that when it is a de- 

■ r *y • 

creasing function, -j- is negative. 
& ' dx & 

Thus the sign of /' (x) for any value of x is positive or 
negative according as/(.r) is, for that value of x, an increas- 
ing or a decreasing function. For example, it is evident that 
the value of the derivative of sin a- must be positive when x 
is between o and \ti, negative when x is between \rt and J-tt, 
and so on. 

When the notation — r - is used, the value of the derivative 
ax 

corresponding to a particular value a of x is expressed by 
-4- which is equivalent tof'(a). See Art. 2. 

CiX /m 



Examples III. 

1. If a point move in the straight line 2y — *jx — 5 = o, so that its 
ordinate decreases at the rate of 3 units per second, at what rate is the 
point moving in the direction of the axis of xl ■ 

dx _ _6 
~dt ~ 7 " 

2. If a point starting from (o, b) move so that the rates of its co- 
ordinates are k and k', show that its path \sy=mx + b, m being 

equal to t- 

Express x and y in terms oft {Art. 14), and eliminate t. 

3. If a point moving in a curve passes through the point (5, 3) 



22 FUNCTIONS RATES AND DERIVATIVES. [Ex. III. 

moving at equal rates upward and toward the left, find the value of 

— ^ , also the equation of the tangent line to the curve at the given 

dx_\ b 

dy~\ 
point. -^ I = - i,andj + .r=8. 

4. If a point is moving in the straight line 

x cos a + y sin a = ft, 

its rate in the positive direction of the axis of x being / sin a, what is its 

rate of motion in the direction of the axis of y? 

— / cos cc. 

5. Given ay sin a — ax + ax cos a — & sec a = o; show that $ is con- 
stant and equal to \a. 



6. If/(,r) = tan x, show that/'6r) must always be positive. 

7. Sh 
negative 



7. Show, by tracing the curve, that if y = x % , — Z can never be 



CHAPTER II. 

The Differentiation of Algebraic Functions. 



IV. 

The Square. 

29. In establishing the formulas for the differentiation of 
the simple algebraic functions of an independent variable, we 
find it convenient to begin with the square. The object of 
this article is, therefore, to express d(x*) in terms of x and 
dx. 

We first deduce a relation between two values of the de- 
rivative of the function and the corresponding values of the 
independent variable ; for this purpose, we assume two values 
of the variable having a constant ratio m. Thus, if 

z = m x, z" 1 = ni~ x*. 

Differentiating by equation (2), Art. 21, 

dz = m dx, and d(z') = ;/z a d(x a ) ; 

dividing, we obtain 

d { S) d(x 2 ) 

—j — = m — = — 
dz ax 

Whence, dividing by z — m x to eliminate ;«, we have 



1 d(z-) _ 1 d(x*) 
z ' dz x dx 



CO 



24 ALGEBRAIC FUNCTIONS. [Art. 29. 

The derivatives —. — and — y— are, by Art. 23, functions 

of z and of x respectively, independent of the values of dz 
and dx\ moreover, equation (1) is true for all values of x 
and z, these quantities being entirely independent of each 
other, since the arbitrary ratio m has been eliminated. There- 
fore, either of these quantities may be assumed to have a 
fixed value, while the other is variable ; hence it follows 
that the value of each member of this equation must be a 
fixed quantity, independent of the value of x or of z. Denot- 
ing this fixed value by c, we therefore write 

x dx ' 
or d(x 2 ) = cxdx (2) 

30. To determine tJie unknown constant c, we apply this 
result to the identity 

(x + lif — x 2 + 2/1 x + h\ 

Differentiating each member (Art. 22) by equation (2), we have 

c (x + h) d(x + h) = cxdx+ 2J1 dx ; 

since d{x + h) = dx, this equation reduces to 

c Jidx = 2h dx, 
or (c — 2) h dx = o. 

Now, since h and dx are arbitrary quantities, this equation 

gives 

c- 2 ; 

this value of c substituted in (2) gives 

d(x 2 ) = 2x dx (a) 

That is, the differential of the square of a variable equals 
twice the product of the variable and its differential. 



§ IV.] THE EQUAJRE. 25 

31. Employing the derivative notation, this result may 
also be expressed thus : — 

If f{x) = x; f'{x) = 2X. 

This derivative is negative for negative values of x, there- 
fore, for these values, x~ is a decreasing function, as already 
mentioned (Art. 10) in connection with the curve illustrating 
this function. 

Since x and dx are arbitrary, we may substitute for them 
any variable and its differential. Equation (a) therefore en- 
ables us to differentiate the square of any variable whose 
differential is known. Thus, — 

d{$* - 3) 2 = 2(5* - 3) *>dx = 10(5* - 3) dx. 

Again, d(ax~ + b xf = 2{ax' + b x) d(ax~ + bx) 

= 2[a x~ + b x) (2a x + b) dx. 



The Sqzmre Root. 
32- To derive the differential of the square root, we put 

y = V*> 

whence f = x ; 

differentiating by (a), 2y dy = dx, 

dx 
or dy = 

y 2y 

Butj=|/^, .'. d ^ x )=T^x () 

That is, the differential of the square root of a variable is 
equal to the quotient arising from dividing the differential of the 
variable by twice the given square root. 



26 ALGEBRAIC FUNCTIONS. [Art. 32. 

Thus, d[ 4/ (a 2 - *■)] == — f# * 



or, using derivatives, 



dx |/ (# 2 - x 1 ) 



Examples IV. 

1. Differentiate (2x + 3) 2 , and find the numerical value of its rate, 
when x has the value 8, and is decreasing at the rate of 2 units per 
second. 

The differential required is denoted by d[(2x + 3) 2 ], and the rate by 

d[(2x + 3) 2 ] i7 . dx 

7; ; the given rate -7— = — 2. 

*# «/ — 152 units per second. 

2. Find the numerical value of the rate of (x 2 — 2jt) 2 , when x = 3, 
and is increasing at the rate of -§- of one unit per second. 

Differentiate the given expression before substituti7ig. 

12 units per second. 

3. Find the numerical value of the rate of \/{y 2 + x 2 ), when y = 7 

and x = — 7, if y is increasing at the rate of 12 units per second, and 

x at the rate of 4 units per second. 

4 4/2 units per second. 

4. If /(V) = or — \/{x" - a-), find f'{x), and show that /(*) is a 
decreasing function. ' x 

./(■*) = ! 



|/(.r 2 - « 2 ) 



5. Differentiate the identity ( ^Ar + 4/tf) 2 = i+a + 2|/fl^ and 
show that the result is an identity. 



/ (x 2 — 2a x\ 
6. Differentiate if - 2 7 ) . • 

y \a 2 — 2« ^/ 



The constant factor 2 Tn should be separated from the variable 

factor before differe7itiation. 1 x — a 



a/ {a 2 — 2ab) a/(x 2 — 2a x) 



IV.] EXAMPLES. 27 



7. If/(>) = (i + .1 A^) = ~- 



.1 + >j 



8. If /(*) = ^(a 3 + 2^ .i- + 6-.v 2 ), f'{x) 



9. If/(jr) = 4/[^+V(l + *')], /"(*) 



\/{d'+ 2b 1 x + ex*) ' 

[ '' + 1/(1 + -Q] 
2 |/(i + **) 



Rationalize the denominator before differentiating, 

A" y- dy . 

11. Given — 2 + ~r, — 1, express -z— in terms of x, and give the values 
a 1 b~ l dx fe 



dx \ o an ^"J a ' <** ~~ + a \/(a' 2 — x*)' 



dy 
12. Given y- = 4<3.r, express -j— in terms of x, also in terms of y, and 



* 



--JL £-♦£- 



2,7 



give the values of 

dx y 



13. A man is walking on a straight path at the rate of 5 ft. per 
second; how fast is he approaching a point 120 ft. from the path in 
a perpendicular, when he is 50 ft. from the foot of the perpendicular? 

Solution : — 

Let x denote the variable distance of the man from the foot of the 

dx 
perpendicular, so that -77 may denote the known velocity of the man, 

and let a denote the length of the perpendicular (120 ft.); then the 
distance of the man from the point is \?{a} + x 2 ), of which the rate of 
change is denoted by 

d\ i/(a°- + x*)] x dx 



dx 
At the instant considered, x = 50 ft., while a = 120 ft., and —j- = — 5 ft. 



tf(a* +x*) dt ' 

dt 



28 ALGEBRAIC FUNCTIONS. [Ex. IV. 

per second. By substituting these values, we obtain — i-j-f. Hence his 
distance from the point is diminishing (that is, he is approaching it) at 
the rate of i-^f ft. per second. 

14. If the side of an equilateral triangle increase uniformly at the 
rate of 3 ft. per second, at what rate per second is the area increasing, 
when the side is 10 ft. ? 15 4/3 sq. ft. 

15. A stone dropped into still water produces a series of continu- 
ally enlarging concentric circles ; it is required to find the rate per 
second at which the area of one of them is enlarging, when its diame- 
ter is 12 inches, supposing the wave to be then receding from the 
centre at the rate of 3 inches per second. 56 -x sq. inches. 

16. If a circular disk of metal expand by heat so that the area A of 
each of its faces increases at the rate of 0.01 sq. ft. per second, at what 
rate per second is its diameter increasing? 1 

100 t/{jtAy ft " 

17. A man standing on the edge of a wharf is hauling in a rope 
attached to a boat at the rate of 4 ft. per second. The man's hands 
being 9 ft. above the point of attachment of the rope, how fast is the 
boat approaching the wharf when she is at a distance of 12 ft. from it? 

5 ft. per second. 

18. A ladder 25 ft. long reclines against a wall; a man begins to 
pull the lower extremity, which is 7 ft. distant from the bottom of 
the wall, along the ground at the rate of 2 ft. per second ; at what rate 
per second does the other extremity begin to descend along the face 
of the wail ? 7 inches. 

19. One end of a ball of thread is fastened to the top of a pole 35 ft. 
high ; a man holding the ball 5 ft. above the ground moves uniformly 
from the bottom at the rate of five miles an hour, allowing the thread 
to unwind as he advances. What is the man's distance from the pole 
when the thread is unwinding at the rate of one mile per hour? 

I |/6 ft. 

20. A vessel sailing due south at the uniform rate of 8 miles per hour 
is 20 miles north of a vessel sailing due east at the rate of 10 miles an 



§ IV.] EXAMPLES. 29 

hour. At what rate are they separating — {a) at the end of \\ hours? 
Q8) at the end of 2^ hours ? 

Express the distances in terms of the time. (a) $■& miles per hour. 

21. When are the two ships mentioned in the preceding example 
neither receding from nor approaching each other ? 

Put the expressio)i for their rate of separation equal to zero. 

When / = £-$ of an hour. 

22. Derive, by the method employed in Art. 29 to determine the 
differential of the square, the result d [ — ) = — %~* c being an unknown 
constant. 



V. 

The Product. 



33- Let x and y denote any two variables ; in order to 
derive the differential of their product, we express xy by 
means of squares, since we have already obtained a formula 
for the differentiation of the square. From the identity 

(x +y)' 2 = x 2 + 2xy+ y 1 , 
we derive 

xy = i (x+yj - \x 2 - if. 

Differentiating, d{xy) = (x + y) {dx + dy) — x dx — y dy, 

therefore, dixy) = y dx + x dy (V) 

Since x and y denote any variables whatever, and dx and 
dy their differentials, we can substitute for x and y any 
variable expressions, and for dx and dy the corresponding 
differentials. Thus, 

(i+x^xdx 



d\_{\ + x 2 ) i/(a 2 - x 2 )] = \/{d - x 2 )2xdx 

x dx. 



•(«* - **) 

2c? — ix 1 — I 



i/(a' - .r°) 



30 ALGEBRAIC FUNCTIONS. [Art. 34. 

34-. Formula (c) is readily extended to products consist- 
ing of any number of factors. Thus let^ x 2 x z . . . . xp denote 
the product of / variable factors, then 

d(x\ x\ 'x** • • • Xp) = x^ x % • • • Xp dx x + x x d(x 2 x a • • • Xp) 
= x. 2 x 3 • • • Xp dx x -f x x x z - - • x p dx n _ + x x x^ d(x % ■ • • Xp) 
= x 2 x 3 • • • Xp dx x + x t x z • • • Xp dx 2 • • • + x x x„ • • • x P _ x dxp. . (d) 



The Reciprocal. 

35. The differential of the reciprocal may now be 
obtained by means of the implicit form of this function. 
Denoting- the function by y, we have 



1 

y — — . • . xy 



Differentiating the latter equation by formula (V), we obtain 

y dx + x dy = o, 

v dx 
whence dy = ; 



substituting the value of y, 

41—7 « 

Formula (d) enables us to differentiate any fraction of 
which the denominator alone is variable ; thus, 

Ja + b\ , , N dx 

Aim ] = -(*+*) 



a + x/ K J (a+xf 



§ V.] THE QUOTIENT. 3 1 



The Quotient. 

36. By the term quotient, as used in this article, we mean 
a fraction whose numerator and denominator are both 
variable. In deriving its differential, the quotient is re- 
garded as the product of its numerator by the reciprocal 
of its denominator. Thus, applying formulas (c) and (it), 



dx x dv 

4J=---^ w 

It will be noticed that the negative sign belongs to the 
term which contains the differential of the denominator. 

As an illustration of the application of this formula, we 
have 

_ (2x — a 2{x 1 + b) — 2x (2x — a)\ b + a x — x 2 . 

d — = -r- ) = — : j j~ dx = 2 — 7— , — — -,— dx. 

\x- + b) { *' + b) ix-^rby 

Formula (e) is to be used only when both terms of the 
fraction are variable ; for, when the numerator is constant, the 
fraction is equivalent to the product of a constant and the 
reciprocal of a variable, and, when the denominator is 
constant, it is equivalent to the product of a constant by a 

variable factor. Thus, if it be required to differentiate the 

2 2 

fraction' — , the use of formula (e) may be avoided by first 

making the transformation, 

x~ - 1 - a' x a 



ax a x 



32 ALGEBRAIC FUNCTIONS. [Art. 36. 

since, in this form, one term of each fraction is constant. 
Hence, 



fx" 2 + a 2 \ dx a dx 
\ ax } ~ a x 2 



The Power. 

37. To obtain the differential of the power when the 
exponent is a positive integer, suppose each of the variables 
x \ x i x * ' ' ' x p i n formula (c'\ Art. 34, to be replaced by x. 
The first member contains p factors, and the second / terms ; 
the equation therefore reduces to 

d(x p )=px*- 1 dx (1) 

Next, when the exponent is a fraction, let 

p 
y = xi f then y q = x r ; 

differentiating by (1), p and q being positive integers, we have 

qf' x dy—p x p - x dx, 
therefore, dy = — r dx. 

q f 

Substituting the value of y, 



P **-■ s- : P J 



d(xi)=~- —dx = ~x' 1 dx (2) 



Again, when the exponent is negative, we have 

1 

x — ~r> 
x 



§ V.] THE POWER. 33 



Differentiating by formula (d\ Art. 35, we obtain 

and, since m is positive, we have, by (1) or (2), 

.. . mx m ~ ] dx 

d(x - '") = -,- = - m x -""' dx. ... (3) 

Equations (1), (2), and (3) show that, for all values of ;/, 

d[x n )-.= ux n -'dx (/) 

By giving- to n the values 2, -J, and — 1, successively, 
it is readily seen that this more general formula includes 
formulas (a), (b) and (d). 

38. It is frequently advantageous to transform a given 
expression by the use of fractional or negative exponents, 
and employ formula (/) instead of formulas (b) and (d). 
Thus, 



and d 



d \_(a*-2xy 

L \/{a + xy 



= d{a 2 — 2x 2 )- 2 = 8 (a 2 - 2x 2 )~ 5 xdx, 
= d(a + x) ~ l = — %(a 4- x) ~*dx. 



When the derivative of a function is required, it may be 
written at once instead of first writing the differential, since 
the former differs from the latter only in the omission of 
the factor dx, which must necessarily occur in every term. 
Thus, given 



we derive -/ = (1 +;rW— lx(\ + x*)~*. 2x — T ~ — *r % . 

dx V ' Z V J (I + X )a 



34 ALGEBRAIC FUNCTIONS. [Ex. V. 



Examples V. 

i. From the identity xy = l(x + jk) 2 — \{x—yf derive the formula foi 
differentiating the product. 

a + bx + cj 2 
2. Differentiate . 



a l a \ 
Put the expression in the form — + b + ex. \e ^\dx. 



3. Find the derivative of 

y = ^r—r- See remark, Art. 35. %. = (« 2 - ^ 2 ) ( ^!!^ )S . 

<fr _ 3-r 2 

.</.*• 2 4/(^r 3 — a 3 )' 



4. y= ^(x 3 - a 3 ). 



2x i dy_ _ 4Jtr 3 (2rt; 2 — x*) 

dx ~ (a 2 - xy 



5- y — a i _ x i • Jy— f/z 2 _ r 2 ^ 2 



6. j = (1 + 2x' 1 )(i + 4x 3 ). -j^ = 4x(i + 3x+ ior 3 ). 



7 .y= (a* + xW + 3-^ 2 ). ^ = 3 (5* 3 + ^ + 2a s )x. 

S.y = (i+ X)* (I + x")\ -£= 4(1 + -^) 3 (I + -0(1 + * + 2* 2 ). 






9. j = (1 + x m y + (1 + -*•*)'». 



£- — mn[(i + x>y~Kv" 1 - 1 + (1 + x n y n ~ l x n ~ 1 ]. 



-^ - 1 + 



a — X dy__ _ a J r X 

Xl -y= ~yx-' ~d*~ 2x§ ' 



§ V.] XMPLES. 35 

_ V(x* - «■) dy a>_ 

" y ~ x dx ~~ x* j/{x % — a*)' 

"b c . . dy ab 2x a — a* 



rx V \x' — a-y ' J - dx c x \x* _ a *)l * 

.5. j = +*)•(!-*)■ |= 2 -^£y 

16. ^= + .r) 3 (/> — x)*x\ 

dy 

-+ =x(a + xy (b - xy [2a b + ($b — 6a) x - gx*]. 

dy 2// x* ~ 1 



X n -f I 




t ' y x n — 1 




18. y— (3^+ 2«,r)i (b- 


- rt;r) 


a- — & 

in 1/ - — 





dfr (.r»— if 



dx 



= — $a*x 4/(36 + 2rt.r). 



(2«a- — .r 2 )* 
Pw/ in the form {a 2 — b' 2 ){2a x — x 2 ) ~ K j- = $(a* — b n -) 



x 
20. y = — 



^/{d 1 - x 2 ) 
bx 



21. y = 



\/{2a x — x 2 ) 
/I + X 



X 

23. y = 



:$(a 


2 W * 


— a 


(2a x 


- x 2 )i ' 




dy 


a 2 




dx ( a 2 


— x 2 )l * 




dy a 


bx 




dx~ (2a x 


- x 2 f ' 


dy 


1 





dx (1 — x)\t(i — -r 2 )' 



*/<> 2 + -*" 2 ) - -r 

, , . dy l F a 2 + 2x 2 "1 

Rationalize the denominator. -— = --3- ,. ■ 2 - — 5r + 2x \. 

dx a 2 [_ ^/(rt 2 + -r 2 ) 



36 



ALGEBRAIC FUNCTIONS. 



[Ex. V. 






^" J . /( T 1-3V 



25. y = 



26/ v = 



27. y = 



28. _y = 



29./ 



30- )' 



v> y 



(I 


- x*)% ' 




(1 


1 




(a 


2X 2 — I 


X) n 



x^(i +x*y 

4/(1 + -y 2 ) + y(i - ^r 2 ) 

X 

/ 1 —x 2 

V (FT"-^ 2 ?' 

x (a 2 + x 2 ) /^{a 1 — x'). 



32. y = 



(1 + *«)" 



33- J = 



4/(1 + ^ 2 ) 



^_3 



<£r 



_ 3^r 2 / 2 — x* 
~T'V Ji^x^jf' 

dy 3-y 2 

dx ~ (1 +"x)" + 1 ' 
^ 11a + mb + {m + ;z).r 



34- J 



, jr . See Example 23. 

x + Y(i + x 1 ) 



35- y = 



36. J 



37. ;,= 



|/(r + -i") + j/(i--y 2 ) 
4/(1 + -r 2 )- 4/(1 -* 2 ) 



\/{x 2 + a 2 ) — a 

x \/ (a 4- x) 
\/a — \/{a — x) 



dx 


(a + x) m + l {b + x)»+ l 








dy 1 + 4X 2 


• 




dx ~ x 2 (1 + -r 2 )* 


dy 


4/(1 + x 2 ) + 4/(1 - X 2 


) 


dx 


x 2 4/(1 — X*) 




dy 

dx ~ ( 


2.r 3 — 4_r 


■• 




! _ ^)i (! + , r 2)| 




<^j/ rt 4 + # 2 .r 2 — 4_r* 




^ ~~ \/(a 2 — x 2 ) 






^j/ 2«.r 2n_1 


* 




dx ~ {i+x 2 ) v + 






dy 1 + x 


'• 




d*~ (1 + x 2 f 


23. 


dy 4X* + 3X 2 g 
^r~~ VC**-!- 1) AXm 


dy 


2 


1 




dx ' 


L 1 + 4/(1 - *)J 




dx 


a [ a 


dy 


\/a 


■*" 





^r 2 4/O + *) |/(« 2 — x 2 ) 



§ V.] EXAMPLES. 37 

38. Two locomotives are moving along two straight lines of railway 
which intersect at an angle of 6o° ; one is approaching the intersection 
at the rate of 25 miles an hour, and the other is receding from it at the 
rate of 30 miles an hour; find the rate per hour at which they are 
separating from each other when each is 10 miles from the intersection. 

2£ miles. 

39. A street-crossing is 10 ft. from a street-lamp situated directly 
above the curbstone, which is 60 ft. from the vertical walls of the 
opposite buildings. If a man is walking across to the opposite side of 
the street at the rate of 4 miles an hour, at what rate per hour does 
his shadow move upon the walls — (a) when he is 5 ft. from the curb- 
stone ? (j3) when he is 20 ft. from the curbstone ? 

(a) 96 miles ; (/3) 6 miles. 

40. Assuming the volume of a tree to be proportional to the cube 
of its diameter, and that the latter increases uniformly ; find the ratio 
of the rate of its volume when the diameter is 6 inches to the rate 
when the diameter *is 3 ft. ^. 

41. If an ingot of silver in the form of a parallelopiped expand 
tsVo P art °f each of its linear dimensions for each degree of tempera- 
ture, at what rate per degree of temperature is its volume increasing 
when the sides are respectively 2, 3, and 6 inches ? 

If x denote a sz'de, dx may be assumed to denote the rate per degree oj 
temperature. -//o of a cubic inch. 

42. Prove generally that, if the coefficient of expansion of each 
linear dimension of a solid is k, its coefficient of expansion in volume 
is ^k. 

Solution : — 

Let x denote any side ; then, if V denote the volume, we shall have 
V — ex 3 ; c being a constant dependent on the shape of the body. 

Therefore dV = yx' 1 dx ; 

or, since dx = kx, 

dV ' = 2>k c x 5 = $k V. 

43. Wine is poured into a conical glass 3 inches in height at a 
uniform rate, filling the glass in 8 seconds. At what rate is the surface 



3* 



ALGEBRAIC FUNCTIONS. 



[Ex. V. 



rising at the end of i second? At what rate when the surface 
reaches the brim ? 

Solution : — 

Let h denote the height of the glass (3 inches), Vi its entire 
volume, v and x the volume and height corresponding to the time t t 
and a the time required to fill the glass (8 seconds) ; then 





X 3 


V 


and 


V 


t 
~ a 


therefore 




X = 


ifr h 1 




whence 




dx 

dt ~ 


h 2 

3« 3 




dx~ 


h 






dx~ 




Hi 


t ~3* h 


= h 


and 


~dt 


s 



fl 
\iaS 



44. A person walking along a bridge at the rate of 3 miles per hour 
is 20 yards above, and vertically over, another in a boat, which is gliding 
from under the arch at the rate of 8 miles per hour in a course at right 
angles to the bridge ; at what rate per hour are they separating at the 
end of 3 minutes? 8-541 miles. 



CHAPTER III. 

The Differentiation of Transcendental Functions: 



VI. 

The Logarithmic Function. 



<b 



39. In this chapter, the formulas for the differentiation of 
the simple transcendental functions are to be established. 

We begin by deducing the differential of the logarithmic 
function, employing the method exemplified in Art. 29. 

The symbol log.*- is used in this article to denote the loga- 
rithm of x to any base, and \og b x is used when we wish to 
designate a particular base b. 

Let z=.mx, . • . log 2 = log m + log x, 

differentiating by Art. 21, 

dz = m dx, and d(\ogz) = d(logx) ; 

d(logz) d(\ogx) 
whence — — >- — = 3-^- 

az m ax 

Multiplying by z = mx, to eliminate m, we obtain 

d(\ogz) _ d(\ogx) 
Z —dz— X -llx— (I) 

_,,.., ^(losr z) _ d (log x) _ . 

The derivatives, and , — -, are, by Art. 23, tunc- 



40 TRANSCENDENTAL FUNCTIONS. [Art. 39. 

tions of z and of x respectively, independent of the values of 
dz and dx\ moreover, equation (1) is true for all values of 
x and z, these quantities being- entirely independent of each 
other, since the arbitrary ratio m has been eliminated. Hence, 
in equation (1), one of the quantities, x or z, may be assumed 
to have a fixed value, while the other is variable ; whence it 
follows that the members of this equation have a fixed value 
independent of the values of x and z ; we therefore write 

dilogx) 

x ~ — - = a constant (2) 

dx v 

This constant, although independent of x, may be dependent 
on the value of the base of the system of logarithms under 
consideration. Denoting the base of the system by b, we 
therefore denote the constant by B, and write equation (2) 
thus, — 

sn n Bdx t s 

JQog 6 x) = —^- (3) 



40. To determine the value of B, we establish a relation 
between two values of the base and the corresponding- values 
of this unknown quantity. 

Denoting another value of the base by a, and the corre- 
sponding value of the unknown constant by A, we have 

A dx 

Woga*) = — (4) 

The relation sought may now be obtained by differentiat- 
ing, by means of (3) and (4), the dentical equation 

log a x = log J log,** (5) 

* This identity is most readily obtained thus, — by definition 

7 log-, x 



1 



£ VI.] THE LOGARITHM. 4 1 



thus obtaining 


A dx , ,Bdx 
—--log,,'''—, 


or 


B\og a b = A, 


hence 


logJ* = A, 



that is, A is the logarithm to the base a of b B ; whence we 
have 

b B = a A (6) 

Now, it is obvious that the value of a A cannot depend 
upon b, hence equation (6) shows that the value of b B likewise 
cannot depend upon b\ b B must, therefore, have a value 
entirely independent of b. Denoting this constant value by e, 
we write 

** = « (?) 

Adopting this constant as a base, and taking the loga- 
rithms of each member of equation (7), we have 



Blog E b == 1, 
whence B — 



'fee' 

I 



log £ £' 
Introducing this value of B in equation (3), we obtain 

dx 

< l °s^ = i^j^ ••'■ ^ 

In this equation, the differential of a logarithm to any 
given base is expressed by the aid of the unknown constant e. 

41. The constant e is employed as the base of a system of 

taking the logarithm to the base a of each member, we have 

\0g a X — \og b X log a £. 



42 TRANSCENDENTAL FUNCTIONS. [ART. 41. 

logarithms, sometimes called natural or hyperbolic, but more 
commonly Napierian logarithms, from the name of the in- 
ventor of logarithms. Hence e is known as the Napierian 
base. 

Putting b = e in formula (g) we derive 

d{\og t x) = ^- ,{g') 

The logarithms employed in analytical investigations are 
almost exclusively Napierian. Whenever it is necessary, for 
the purpose of obtaining numerical results, these logarithms 
may be expressed in terms of the common tabular logarithms 
by means of the formula, 

which is derived from equation (5), Art. 40, by writing 10 for 
a and s for b. The value of the constant log, £ will be com- 
puted in a subsequent chapter. 

Hereafter, whenever the symbol log is employed without 
the subscript, log e is to be understood. 



The Logarithmic Ctcrve. 
42- The curve, corresponding to the equation 

y = ^og £ x (1) 

is called the logarithmic curve. 

The shape of this curve is indi- 
cated in Fig. 4. It passes through 
the point A whose coordinates are 
x (1, o), since 

log 1 = o. 

Since we have, from formula (g'), 




§VL] LOGARITHMIC DIFFERENTIATION. 43 

dy i . . 

**+ = £= * ( 2 ) 

the value of tan</> at the point A is unity, and therefore the 
tangent line at this point cuts the axis of x at an angle of 45 , 
as in the diagram. We have from equation (2), 



when 


x > 1 


tan</> < 1, 


and when 


x< 1 


tan0 > 1 ; 



the curve, therefore, lies below this tangent, as shown in 
Fig. 4. 

The point (e, 1) is a point of the curve; let B y Fig. 4, be 
this point, then OR will represent the Napierian base, and 
BR — 1. Since 

OA - 1, and AR > BR, 

OR > 2; 

that is, the Napierian base e is somewhat greater than 2. 

The quantity e is incommensurable : the method of com- 
puting its value to any required degree of accuracy is given 
in a subsequent chapter. 

Logarithmic Differentiation. 

43. The differential of the Napierian logarithm of the 

dx 
variable x y that is the expression — , is called the loga- 
rithmic differential of x. 

When x has a negative value, the expression log^r has no 
real value ; in this case, however, log (— x) is real, and we 
have 

d(—x) dx 
d[\og (-*)] = ^ = — . 



44 l^RANSCENDENTAL FUNCTIONS. [Art. 43. 

This expression therefore, in the case of a negative quantity, 
is identical with the logarithmic differential of the positive 
quantity having the same numerical value. 

44. The process of taking logarithms and differentiating 
the result is called logarithmic differentiation. By means of 
this method, all the formulas for the differentiation of alge- 
braic functions may be derived. 

In the following logarithmic equations, it is to be under- 
stood that that sign is taken in each case which will render 
the logarithm real. 

By differentiating the formulas, — 

log(±xy) = log (+.*-) + log(±y), 
log(±j) = log(±*) -log(±j), 
\og{±x n ) = n \og(±x), 
we obtain 





d(xy) 


dx 

X 


dy 




x y 


y 


I 

X 


<f) 


dx 

X 


dy 

y 




d{x") 
x n 


dx 
= 11 — -. 

X 





These formulas are evidently equivalent to (V), (e), and (/), of 
which we thus have an independent proof. 

45. The method of logarithmic differentiation may fre- 
quently be used with advantage in finding the derivatives of 
complicated algebraic expressions. For example, let us take 

4/(2*) ( 1 - **)» ■ 
u = i (I) 

Hence, we derive 



§VI.] THE EXPONENTIAL FUNCTION. 45 

log" U = i log (2.r) + | log (I - X-) - |- log (x — 2), . . (2) 
differentiating, 



du 1 x I 



7/ d£r 2^ 2 I — „v a 8 ;tr — 2 
adding - and reducing:, 

dfo — 8,r 3 4- 24,r 2 — x — 6 



(3) 



therefore 



udx 6 (1 — ,r 2 ) (# — 2) x ' 

</« — 8_v 3 + 2zpr 2 — ^' — 6 



^ 3 (2xf (I — **)* (* — 2) 1 



In this example, the function u is real only when x is 
positive and less than unity ; hence the factor x — 2 and the 
function u are negative, and we therefore ought strictly to 
write in equation (2) log(— u) and log (2 — x). See Art. 43. 



Exponential Ftcnctions. 

46. An exponential function is an expression in which an 
exponent is a function of the independent variable. The 
quantity affected by the exponent may be constant or vari- 
able. In the first case, let the function be denoted by 

y — a* (1) 

If a is negative, a* cannot denote a continuously varying 
quantity. We therefore exclude the case in which a has a 
negative value, and regard a* as a continuously varying pos- 
itive quantity. 

Taking Napierian logarithms of both members of equation 
(1), we have 

\ogy = x\oga; 



differentiating by (g'), 



46 TRANSCENDENTAL FUNCTIONS. [Art. 46. 

— = 102: a. ax \ 

y 

hence - dy = loga.ydx, 

or d(a x ) = loga.a* dx (Ji) 

Exponential functions of the form e* are of frequent occur- 
rence. Putting a — e in formula (/i), we have 

d(e x ) = c*dx\ . (h!) 

hence the derivative of the function e x is identical with the 
function itself. This function is the inverse of the Napierian 
logarithm ; it has been proposed to denote it by the symbol 
exp^r. 

47. When both the exponent and the quantity affected by 
it are variable, the method of logarithmic differentiation may 
be employed. Thus, if the given function be 

z = (nx)* 2 , 
we shall have logz = x* log (11 x) ; 

dz dx 

differentiating, — = x* — + 2x log {nx)dx 9 

z x 

hence • d[(n x) x *~\ = (n x) x * x[i + 2 log (n x)~\ dx. 

Examples VI. 



1. Given the function y ~ loafjjr; show that -=— = — ^~ 

dx_\ £ e 



and 



hence prove that the tangent to the corresponding curve, at the point 



whose abscissa is e, passes through the origin. 
Put a ; = x = e in equation 5, Art. 40.. 



§ VI.] EXAMPLES. 47 

2. y = x n log x. ~ — x n ~ x (I + 11 log x). 



dx 



3- y = log (log :r). 

4. y = \og[\og(a + bx n )]. 

5. y = tfx — log ( */x + 1). 



6. j = log-*-- 





dy _ I 




dx .rlog.r' 


4y . 


nbx n ~ x 


dx 


(a + fix") log (a + dx*)' 




dy I 




dx 2 ( t/x + 1)' 




</y |/« 



\/a — \/x dx {a — x) \/x 

Put in the form, log ( \/a + \/x) — log ( \/a — \fx). 

7 .y = log [ tf(x - a) + tf(x - b)}. --- 



dx 2 tf[(x — a) (x — £)] 

dy _ 1 

~Zr ~~ \/(x 2 ± a 2 )' 

x dy _ 1 

4/(1 +~^)' ^p ~ x{i + x 2 )' 

4/(1 + x) 4- V(i - x ) dy i_ 

4/(1 + *) — 4/(1 — -r)' ^ jr 4/(1 — jit 2 )" 



S.y = \og[x+ V(.r 2 ±« 2 )]. 
9. ^ = log 
10. y = log 



i r . // 2 a\i dy 4/O 2 — ^r 2 )v— jr 

11. y = log .[, + ^ - **)]■ ^ = ^rfpypqi V-^)] - 

12. J = log— - r — ^ — - • -£ = - + 



*J{x- + a 2 ) -^ x dx x \/{x 2 + a 2 ) 

dy 1 1 



& y = log [ 4/(1 +* 2 ) + ^(i-* 2 )]. 



dx x x 4/(1 — ^ 4 ) 



. , . « (2^ — «) ^/ x 2 + a 2 



48 TRANSCENDENTAL FUNCTIONS. [Ex. VI. 

15. y = a x2 . -^-=2loga.a x2 x. 

1 

16. y = £i +*. 



17.^ = ^(1 _*■■). 



18. j/ = (* — 3) e 2 * + 4X e*. -^- = (2X — 5) £ 2r + 4(jt + l)e*. 

__ £ x — £ ~ x dy _ 4 

20. y = b a \ ^=\ogaAogt.& aX .a*. 



23. _y = log (e x + e-*). 



24. y = a ] °z x . 



27. y = x l °z x . 



dy _ 




I 


1 


<for 


(1 


+ X) 2 






: F?(l 


- 3 X 2 


-X 3 ) 



dy _ 

~dx~~ 

x dy _ e x (i — x)— I 

"^7* dx ~ (e* — i)* 2 

dy _ e x — e~ x 



21. y =^ ra . -v- = na xn . x n ~ l . loga. 

ax 



22. y = 



<£f x 



. e dy 1 

2 5 .j = iog r _. ^-rr^"* 

26. j =*•. ~dx~ =xX ^ 1 + log ^' 






^_ 



28. / = e < ^ = e*". * x (1 + log*). 



§ VI.] EXAMPLES. 49 



i 
29. y = x x , 



^L = J I -log* 

dx ' ' x 1 ' 



cX dy ^ 1 + x loef.r 

J -^ dx x 

32. j = a a (a- log a — 1). -^- = (log «) 9 .r a*. 



33. y — 2e Vx (x% — 3-T + 6-T* — 6). 



-£- =z X£ V 

dx 



34- y 



(x — i) § dy_ _ _ (x— i)g(7.r 2 + 30^ — 97) 

(_ r _ 2 f (x - 3) 1 ' dx 12 (jr — 2)* (.r - 3)"" 



5^ Art. 45. 



35- J 



36. / = 



37- y 



\/[ax{x — 3^)] ^ |/«(;r 2 — 8rt,r + I2<? 2 ) 

d* 2[x{x - 3 a)f( X - Aa f 

dy _ x- (x + 3)* 
dx ~ ( X + 2 ) 5 (,r+ 1)* 

<Ck _ C* — 2 ) s (- r ~ — i- x + 1) 

^~ C*-i) f C*--3) V 



|/Gr - 4«) 




(x + 1)* (.r + 3) f 




C* 4- 2j 4 




(* - 2) 9 




(. r _ I )*(. r _ 3 )V- 




(or a — 2X + 2)1 (V + 


1 



3 8 - y = 





y~- 


exVtf-t 


) 


39' 


~x+ S /(x*~ 
-( 


-I)' 
V 


4 U 


\i + V(*- 


-**))' 



dy _ (Sx z —2\x~ + 26x —1 $)(x + 1)* 

^' r 2 (X- —2X + 2)*{X + 1)5 

dy _ 2 \/(x"— i)«« v <» 1 - , > ! 



5o 



TRANSCENDENTAL FUNCTIONS. [Ex. VI. 



■ _ x / x \ n 

4I - u - V (l-X^[l + V (l-X^) • 

du _ / x y i + n 4/(1 — x*) 

~d* ~ \ l + V( l -x*) ) (1 - x 2 Y 

Put [ rr ] = y, and use the result obtained in Ex. 40. 



VII. 



The Trigonometric or Circular Functions. 

48 > In deriving the differentials of the trigonometric 
functions of a variable angle, we emplo}^ the circular measure 
of the angle, and denote it by 6. Thus, let s denote the 
length of the arc subtending the angle in the circle whose 
radius is a, then 

e= s -. 



In Fig. 5, let A be a fixed line, and OP an equal line 
rotating about the origin O ; then P 
will describe the circle whose equation 
(the coordinates being rectangular) is 

x~ +y = a 1 . 

The velocity of the points is the rate of 

s, and (see Art. 17) is denoted by -7-, 

which has a positive value when P 
moves so as to increase 6. Let PP, 
taken in the direction of the motion of P, represent ds; then, 
according to the definition given in Art. 25, PP' is a tangent 
line, and PB and B P will represent dx and dy, as in Art. 26. 




Fig. 5. 



§ VII.] THE SINE AND THE COSINE. 5 1 

49. We have first to show that the line PP, which is a 
tangent to the curve according to the general definition (Art. 
25), is perpendicular to the radius. 

Differentiating the equation of the circle, we have 





xdx+ydy = 0; 


whence 


dy x 

tan = ~~ = . 

ax y 


Now (see Fig. 5), 


£ = tan 0, 

X 


therefore, tan0 


= — cot 6 = tan (d ± \ n) y 


or, 


<p = d±i7t; 



hence the tangent line is perpendicular to the radius. 
Assuming <f> to be the angle between the positive directions 
of x and ds, we have 



The Sine and the Cosine. 
50. From Fig. 5, it is evident that 



v x 

sin(9r=-, and cos<9 = -; 

a a 

therefore <^(sin^) = — -, and <^(cos0) = --. . . . (1) 



In equations (1) we have to express dy and dx in terms of 
6 and dd. 



52 TRANSCENDENTAL FUNCTIONS. [Art. 50. 

Again, from the figure, we have 

dy — sin (p. ds, and dx — cos <f>. ds ;* 

substituting in equations (1), we obtain 

■>- . . ds 1 , f n ^ - / \ 

<tf(sin0) = sin0— , and d(cosO) = cos</>— . ... (2) 

_$■ 

Since = + £ n, and - = 0, 

i ds 7 

sin<£ = cos0, cos0=— sin0, and — = ad. 

Substituting these values in equations (2), we obtain 

d(sind) = cos Odd, (z) 

and d(cos6)= — sinddO (J) 

The Tangent and the Cotangent. 

51. The differential of tan0 is found by applying formula 
(e) to the equation 

sin# 



tan = 



cosO' 



cos d(sin 6) — sin 6 d(cos 6) 

thus, ditand) = rs L > 

K J cos 

J a 

or */(tan0) = — ^ = sec* Odd (£) 



*In Fig. 5, e£c is negative ; but, <p being in the second quadrant, cos $ is 
likewise negative. 



§VIL] THE TANGENT, SECANT, ETC, 53 

The differential of cot is found by applying formula (k) 
to the equation 

cot 6 = tan {^n — 0) ; 
whence d(cotO) = — - = — cosec" Odd. ...(/) 



The Secant and the Cosecant. 

52. The differential of sec is found by applying formula 
(d) to the equation 

sec0 



cosfl' 

sin 0^0 _ . . 

whence ai sec d) = :,-— - = sec 6 tan 6 ad. . . (m ) 

v ; COS"0 v ' 

The differential of cosec 6 is found by applying formula 
(m) to the equation 

cosec 6 — sec (-£- n — 0) ; 



cos Odd , , . 

whence ^(cosec0) = ^tt- = — cosec cot #;/0. . (n) 



The Versed-Sine. 
53. The versed-sine is defined by the equation 
vers 6 = I -^ cosfl; 
therefore ^(vers 0) — sin */0 (o) 



54 TRANSCENDENTAL FUNCTIONS. [Ex. VII. 



Examples VII. 

i. The value of <tf(sin0) being given, derive that of d(cosO) from 
the formula 

cos0 = sin (i7r — 6) ; 
also from the identity 

cos 2 6 =i — sin 2 0. 

2. From the identity sec 2 = i + tan 2 6, derive the differential of 
sec0. 

3. From the identity sin 2 = 2 sin cos B, derive another by taking 
derivatives. cos 2 = cos 2 — sin 2 0. 

4. From the identity sin (0 + £77-) = £4/2 (sin0 ± cos0), derive an- 
other by taking derivatives. cos ( Q ± ^tt) =1^/2 (cos0 T sin0). 

5. Prove the formulas : — 

aT(log sin 0) = — ^(log cosec 0) = cot dB ; 
d(\og cos 0) = — <^(log sec 0) = — tan dB ; 
dQog tan 0) = — dQog cot 0) = (tan + cot 0) dB. 

6. Obtain an identity by taking derivatives of both members of the 

equation 

, 1 — cos 

tan 1 = 7— r — . 

sin0 

1 — cosG 



i sec 2 i B 



dy 

7. y = 6 -f sin cos 0. ~JB~ 2 C03S 

8. j = sin — |sin 3 0. -tq = cos 8 0. 

sin© -^ 1 + cos 2 

9 ' ^ = ^(cos~0)' ^ = 7(cos~0)* ' 



§ VII.] EXAMPLES. 55 

dy 
10. y = i tan 3 — tan + 0. -£- = tan 4 0. 

ii. y = ^tan 3 + tan 0. — = sec 4 6. 

du 



12. y = sine*. -=- =. e* COS e*. 



-7- = sin x- + 2-r 2 cos x 2 . 
dx 

14. _y = # s,n *. -,— = log^.a 6inj: cos^. 



15. j = tairO + log (cos 2 5). 

16. y = log (tan + sec 5). 

17. j = logtan(irr + £0). 

18. y = x + log cos (i^ — ;r). 

19. y = log |/(sin .r) + log ^/(cos ;r). 



dO "" 


2tan 3 0. 


dy 

de 


= sec Q. 


dy 
dO 


1 

~~ COS 9* 


dy 


2 


d ~ 1 - 


f tan x 


dy 
dx~ 


cot 2X. 



dy 
20, j/ = sin n 9 (sin £) n . d9~ n ( sin 6 ) n " ' sin ^ + *) 9 * 

^ cos 3 .r — sin 3 x 
dx ~ (sin x + cos x) 2 ' 

dy 

-j- = e?* (a cos bx — b sin <5x). 



sin x 




* 1 + tan .r 




22. y = e^cos £;r. 




. /acosx 


— bsinx 



dy _ — ab 

dx ~ a 2 cos 2 .r — b 2 sin 2 ;r' 



$6 TRANSCENDENTAL FUNCTIONS. [Ex. VII. 



±- 


— 


£ cc sec 2 £ a; 


dx 


.r 2 


i-> 


+ 


i)£ ax sin^r. 



1 
24. y = tan e* 



25. _y = e a * (<z sin^r — cos x). 



dy 

26. y = e x (cos .r — sin x). -7- = — 2 e* sin ^r. 

27. _y = e— a2j;2 cos r .r. -r- = — e — a2a:2 (2a 2 jrcosrj + rsinrjr). 

(sinnx) m dy m7i(s\nnx) m — l cos(mx — nx) 

28. y = 



(cos;/z.r) M * dlr (cos;«^) n+1 

29. y = tan 4/(1 — jt). 



^j/ _ — [sec |/(i — .r)] 2 
dkr ~~ 2 |7(i — jjt) "0 



*#> / , sin.r\ 

30. y = .r sina; . -v- = x unx cos;r.log-r + I. 

dy cos (log ^^r) 

31. j = sin (log nx). j- = -. 

dy 

32. j = sin (sin ;r). ~r~ = cos x. cos (sin x). 

2 3COS^r jt dy 2^ 

33- -^ ~~ sin 2 ^-cosx~ sin 2 ;r ^ °^ an 2' atr ~~ sin 3 .r cos 2 x ' 

34. Given x — r cos 0, and y — r sin 6, prove that 

^KsinQ + rfUrcosQ = tf'r, 
and ^ycosQ — Ursine = rdB. 

35. From x = ^cosfi, and y — rsinQ, deduce 

(dx) 2 + {dyf = (dr)' + r\dB)\ 

36. The crank of a small steam-engine is 1 foot in length, and 



§ VII.] EXAMPLES. 57 



revolves uniformly at the rate of two turns per second, the connect- 
ing rod being 5 ft. in length ; find the velocity per second of the 
piston when the crank makes an angle of 45 with the line of motion 
of the piston-rod; also when the angle is 135 , and when it is 90 . 

Solution : — 

Let a, b, and x denote respectively the crank, the connecting-rod, 
and the variable side of the triangle ; and let denote the angle be- 
tween a and x. 

We easily deduce 

x = a cos Q + \t(b" — a~ sin 2 0) ; 

. dx ! , tf 2 sinOcosQ \ dO 

whence -— = — i«sin0 + 

at 

In this case, -zj = 4?r, a = 1, and b = 5 



dt \ ^{b' 1 — d-sm 2 v)J dt 

dt 



When e = 45°~ = --^ft. 

dt 7 

37. An elliptical cam revolves at the rate of two turns per second 
about a horizontal axis passing through one of the foci, and gives a 
reciprocating motion to a bar moving in vertical guides in a line with 
the centre of rotation : denoting by the angle between the vertical 
and the major axis, find the velocity per second with which the- bar is 
moving when = 6o°, the eccentricity of the ellipse being i, and the 
semi-major axis 9 inches. Also find the velocity when B = 90 . 

The relation between 6 and the radius vector is expressed by the equation 

a (1- e') 
r = 



^ CO3 

dr 
When = 6o°, — - = — 12 4/3 tt inches. 

38. Find an expression in terms of its azimuth for the rate at which 
the altitude of a star is increasing. 

Solution : — 

Let h denote the altitude and A the azimuth of the star,/ its polar 
distance, t the hour angle, and L the latitude of the observer ; the 
formulas of spherical trigonometry give 

sin/£ = sin L cosfi + cosZ sin/ cos /, . . . . (1) 
and sin/ sin / = sin A cos h (2) 



58 TRANSCENDENTAL FUNCTIONS. [Ex. VII. 

Differentiating if), ft and L being constant, 

cos k-jj = — cos L sm/ sin t, 

whence, substituting the value of sin ft sin/, from equation (2), 

dh r • a 

-—= — cos L sin ^4. 

«/ 

It follows that 37 is greatest when sin A is numerically greatest ; that 

is, when the star is on the prime vertical. In the case of a star that 
never reaches the prime vertical, the rate is greatest when A is greatest. 

39. Trace the curves y = sin x,y = tan x, and y = seer, determining 
in each case the value of tan <p, for the point at which the curve cuts 
the axis of y. 



VIII. 

The Inverse Circular Functions. 

54. It is shown in Trigonometry that, if 

x = sin 6, 
the expressions 

2H7t + d and (2n+i)it — t . . (1) 

in which 11 denotes zero or any integer, include all the arcs 
of which the sine is x ; hence each of these arcs is a value 
of the inverse function 

sin "" 1 x. 

Among these values, there is always one, and only one, 
which falls between — Jtt and -\-\n\ since, while the arc 



§ VIII.] INVERSE CIRCULAR FUNCTIONS. 59 

passes from the former of these values to the latter, the sine 
passes from — i to + 1 ; that is, it passes once through 
all its possible values. 

Let 0, in the expressions (i), denote this value, which we 
shall call the primary value of the function. 

55. In a similar manner, if 

x = cos 0, 

each of the arcs included in the expression 

211 7t ± (2) 

is a value of the inverse function 



One of these values, and only one, falls between oand tt ; 
since, while the arc passes from the former of these values 
to the latter, its cosine passes from + 1 to — 1 ; that is, once 
through all its possible values. In expression (2), let denote 
this value, which we shall call the primary value of this 
function. 

56- In the case of the function 

cosec -1 ^ - , 

the definition of the primary value that, was adopted in the 
case of sin~ 1 .r, and the same general expressions (1) for the 
values of the function, are applicable. 
In the case of the function 

sec"" 1 -*-, 

the definition of the primary value adopted in the case of 



60 TRANSCENDENTAL FUNCTIONS. [Art. 56. 

cos -1 .*- and expression (2) for the general value of the 
function are applicable. 

Finally, in the case of each of the functions 

tan -1 .* and cot -1 x 

the primary value (6) is taken between— \n and + -§-#, and 
^ the general expression for the value of the function is 

nrf+d (3) 

The Inverse Sine and the Inverse Cosine. 
57. To find the differential of the inverse sine, let 





6 = sin ~ 1 x; 


then 


x = sin 6, and dx = cos Odd, 


or 


dx 
dd= -. 

COS 6 


Now, 


cos (9 = ± |/(i — sin 2 0) = ± 4/(1 — x*), 


hence 


d(sm- l x)= - 



(I) 



If denotes the primary value of this function; that is, the 

value between — ^tt and +J^, cosfl is positive. Hence the 

upper sign in this ambiguous result belongs to the differential 

of the primary value of the function ; it is therefore usual to 

write 

dx 
d(sin-^) = -^—^ {p) 

Since we have, from expressions (i), Art. 54, ' 

d(2n n + d) =r dd, and d\_(2n+ i)tt — 6] == — dd, 



§ VIII.] THE IX VERSE SINE AND IX VERSE TANGENT. 6l 



it is evident that the positive sign in equation (i) belongs not 
only to the differential of the primary value of sin _1 .r, but 
likewise to the differentials of all the values included in 
2mr + 0; and that the negative sign belongs to the differen- 
tials of the values of sin -1 * included in (2/1 -\- 1 ) n — 0. 

58. Similarly, if 

= cos ~~ ' x, x = cos ; 

dx 

whence do — . — - a , 

— sin# 

dx 

If denote the primary value of the function which in 
this case is between o and zr, sin is positive ; hence the up- 
per sign in this ambiguous result belongs to the differential of 
the primary value. It is therefore usual to write 

_ dx 
rf(co S -'*) = — ( — ^ (?) 

Since, from expression (2), Art. 55, Ave have 

d(27i 7t ±6) = ±dd; 

it is evident that the upper and lower signs in equation (1) 
correspond to the upper and lower signs, respectively, in the 
general expression 211 n ± 0. 

The Inverse Tangent and the Inverse Cotangent. 

59. Let 

= tan ~~ * #, then x — tan 6 ; 

differentiating, we derive, 

dx 

dd = 



sec 2 # 



62 « TRANSCENDENTAL FUNCTIONS. [Art. 59. 



But sec 2 0=1+ tan 2 6 = 1 + .r 2 , therefore, 



^(tan- 1 ^)^— 2-j (r) 



No ambiguity arises in the value of the differential of 
this function ; since, from expression (3), Art. 56, we have 

d{n 7T + 6) = dd. 

Similarly, putting 

= cot~ l x, 

dx 
we derive d(cot~ x x) = 5 (s) 



The Inverse Secant and Inverse Cosecant. 
60. Let 

0=sec _1 .ar, then ^ = sec(?; 

differentiating, we derive 

dx 
dd 



sec 6 tan 6 



But sec0 = x, and tanfl = ± |/(sec 2 — 1) = ± ^{x 1 — i) : 
therefore, 

<tf(sec ^r) = -r-=. ;. 

If x is positive, and if denotes the primary value of the 
function, tan0 is positive. Hence it is usual to write 

dr 

d(sec-\v)= /f , W 

x \/{x — I ) 



§ VIII.] THE INVERSE SECANT. 63 

When x is negative, if denotes the primary value of the 
funetion, which in this case is in the second quadrant, tan0 is 
negative ; consequently the radical must be taken with the 
negative sign. Hence, since x is also negative, the value of 
the differential is positive, when the arc is taken in the 
second quadrant. 

In like manner we derive 

dx 
d{cosec-\r) = - xV{x ,_^ («) 

Similar remarks apply also to this differential when x is 
negative. 

The Liver se Versed-Sine. 
61. Let 

= vers _1 .r, then x = versfl = 1 — cos 0, 

and 1— x = cosd, .*. dd=- — -. 

sin0 

But sin0 = |/(i — cos 2 0) = \/{2x— x*), therefore, 

d(vers~ 1 x) = —-T- ^ (v) 

v J \/\2x — x ) 



Illustrative Examples. 

62. It is sometimes advantageous to transform a given 
function before differentiating, by means of one of the 
following formulas : — 

• -i a _, -i a -i0 . -i a <-i8 

sin -75 = cosec - , cos -5 = sec — , tan -75 = cot - . 
p a p a p a 



64 TRANSCENDENTAL FUNCTIONS. [Art. 62. 

1 hus, let y = tan : — , 

^ 1 + e^ sin ^ 

then 7= cot -1 (£ _;r sec^ + tan^). 

By formula (j), 

<^ e - -* sec ;r tan;tr — e - -*" seer -f sec 2 ;tr 



dx ~ sec 2 ;r + 2t~ x sec ^ tan ^ + e - 2 x sec 2 x ' 

multiplying both terms by e 2X cos 2 ^, 

<fj/ e*(cos .*■ — sin .r — e*) 
<3^r — I + 2 s x sin .*• + e 2X 

63. Trigonometric substitutions may sometimes be 
employed with advantage. Thus, let 

y — tan ~ 2 



|/(i+^ 2 )+ 1 
If in this example we put x = tan0, we have 

tan 6 . sin 6 



y = tan ' — — = tan * 



sec# + 1 1 f-cos# 

= tan" 1 (tan £0) =$0 = Jtan" 1 ^ 

Examples VIII. 

I Derive from {£), (; ), and (/) the formulas : — 



J ■ 1 *\ ^ 



§ VIII.] EXAMPLES. 65 

/ x\ adx 

d sec- 1 — = — -7— 2 57' 

\ a) x tf{x 2 - a 2 ) 

2. Derive <y(sec ~ x x) from the equation sec - l x = cos _1 — • 



-t.' 



3. Derive d[ cot -1 — J from the equation cot -1 — = tan -1 — 



4. y = sin -1 (2x-). 



6. y = sin (cos _1 .r). 

7. _y = sin -1 (tan^r). 

8. y = cos -1 (2COS.r). 

9. y = xs'm-' [ x + 4/(1 — x*). 

10. jj/ = tan -1 ^. 

11. y = (.r 2 + 1) tan -1 .r — x. 



x 
12. y = a 3 sin" 1 — + x \/{a 2 — x 2 ). 



13. j = tan~ 


j 7/z.r 


l-.r 2 


14. y = sin~ 


,* + I 



</j> 4-T 



dx 4/(1— 4-r 4 )' 
5. j = sin- 1 (cos^r). ^=~ r 



^ _ .r 

dy sec 2 x 





dx y(i 


— tan 2 


*)' 


dy 




2 sin^r 




dx 


4/(1 


— 4COS 2 


*y 






= sin - 


1 X. 




# 


1 






dx' 


~ e x + e 


— x ' 




dy 
dx~ 


2-rtan - 


l x. 




~dx ~ 


i/O 2 -. 


*■). 



dy _ ?/z(i + x 2 ) 

dx~ 1 + (;;z 2 — 2) .r 2 + -r 4 

</>/_ I 

4/2 *£r — 4/(1— 2.r — ^r 2 ) 



66 TRANSCENDENTAL FUNCTIONS. [Ex. VIII, 

x dy i 

15. y— tan-. 1 



19. y = |/(i — x" 2 ) sin-^r — x. 



21. y = cos -1 



22. j/ = tan- ] j/ — 



x sin — 1 .r 
23. y = 



I — COS^r 



cosx 



3 -^ = 7(^^) +log4/(I ~- r) - 

24. y = (j/+ #) tan -1 i/ \/(ax). 



25. y = tan 

3 ^ 2 + x 



2CX + b 

26. y = tan ~ ] —7 75. 

-> 4/(4^ -£ 2 ) 



27. jj/ = £sin-la! 



4/(1— .r 2 )' <£r 4/(1— jf 2 )' 

^ dy 1 

16. j/= sec- 1 — t-j -^r. 



4/O 2 — x 2 ) ' dx ^/(a- — x z ) ' 

x dy a 

l 7- y = sin- 1 — -. -=- = --3—; — 2* 

V(^ 2 + « 2 ) ^ « + ■* 

18. j =± sin- 1 ^/(sin r). — = £ 4/(1 + cosecr). 



<^j/ xsin —1 x 



^r 4/(1 — .r 2 ) 



m + x dy 

20. _y = tan -1 



1 — m x ' ^r 1 + x 2 ' 

1 — ;r 2 dy 2 







dx 1 + x 2 ' 






dx 




dy 


sin -I x 




dx 


~ (1 - .r 2 )* ' 




dx ~ 


= tap-»|/^ 


dy 




V3 


dx 


~ 2 


O 2 +.* + i)- 


dy 


-ia 


^(4^ — £ 2 ) 


dx~ 


+ bx + <r.r 2 * 




dy 


£ sin — 'a; 



dx Y('~ x ')' 



VIII.] EXAMPLES. 67 



2Q y = X £ tan x . 



dx \ X* + I J 



30. y = e & + *") ta: > ! * ^ = (i + 2.rtan-' x) e^ + * 9 ) t;l " ' 

dx 

,4- 3-*" 9 ^ 3 



31. y = cos -1 



r/.r x |/(.i-' 2 — i)* 



32. y = tan - [x + </(, - .t-)j. - = 3y(| ^, [l+ ^ ( , ,^3 ■ 

,^ + ^COSJT dfr V(rt a — 3 2 ) 

33. y=sin- 1 — —7 ■. -jr = — , -. 

x \/$ dy 1 

T.A y = sec - — — ^ = 

J ^ 2 \/{x* + .1' — 1) * *£r .r 4/(^r- ■+- x — i) " 



35. j = tan 



^K 3<? 



« 3 — 3^,r 2 ' dx a' 2 + x 2 ' 



M*—a*)sinx dy */(? — a~) 

36. y = tan- 1 — -. — . -~ = -j— . 

^ ' a + #cos,r <f.r b + a cos^r 

• _i --f-i/C^ - ^) dy _ \/(a — 6) 

37 ' y ~~~ Sm i/[a(i +:^)\ ' dx = {\Tx ll )~x/{a+bx 1 ~y 

38. Trace the curves y = sin -1 x, y = tan -1 x, and y = sec -1 .r, indi- 
cating the portions which correspond to the primary values of the 
functions. 



IX. 

Differentials of Functions of Two Variables. 

64. The formulas already deduced enable us to differen- 
tiate any function of two variables, expressed by elementary 
functional symbols ; the application of these formulas is, how- 



68 DIFFERENTIATION. [Art. 64. 

ever, sometimes facilitated by a general principle which will 
now be shown to be applicable to such functions. 

The formulas mentioned above involve differential factors 
of the first degree only. It follows, therefore, that the differ- 
entials resulting from their application consist of terms each 
of which contains the first power of the differential of one of 
the variables. In other words, if 

dit = <f>(x,y)dx+ip(x,y)dy (1) 

Now, if y were constant, we should have dy = o, and the 
value of du would reduce to that of the first term in the right- 
hand member of (1); hence this term may be found by differ- 
entiating u on the supposition that y is constant, and in like 
manner the second term can be found by differentiating u on 
the supposition that x is constant. The sum of the results 
thus obtained is therefore the required value of du. 

65. As an example, let 

z = u v . 

Were v constant, we should have for the value of ds, by 
formula (/), Art. 37, 

vu v — x du ; 

and, were u constant, we should have, by formula (h), Art. 46, 

\ogu . u v dv ; 

whence, adding these results, 

dz — u v ~ J (v du + u log 2i dv). 



§ IX.] FUNCTIONS OF TWO VARIABLES. 69 

Although this result has been obtained on the supposition 
that u and v are independent variables, it is evident that any 
two functions of a single variable may be substituted for u 
and v. Thus, if 

u — nx and v = x % , 

we have z = (nx)-**, 

and, on substituting, 

dz = {ii .i')-* 2 - 1 (,r 2 n dx + ?/ .r log (n x) . 2.r dx), 
= x (n x) x ~ [1 + 2 log (nx)]dx, 

which is identical with the expression obtained in Art. 47, for 
the differential of this function. 

The Derivatives of Implicit Functions. 

66. When ;r and y are connected by an equation such that 
y cannot be made an explicit function of x, the value of the 

dy 
derivative -^cannot be expressed in terms of x; it can, how- 
ever, be expressed in terms of x and y, and the numerical 
value of the derivative, for any known simultaneous values 
of x and y, can be determined. Thus, if the given equation is 

x y % — 3 X * y + 6/ + 2x — o, .. . . . . . (1) 

we obtain 

y dx + $xy* dy — 6xy dx — 3^ dy + 1 2y dy + 2dx = o, 

dy j/ 3 — 6xy + 2 f v 

whence -r- — 7 — a r^~\ V 2 J 



70 DIFFERENTIA TION. [Art. 66. 

Now, observing that equation (i) is satisfied by the values, 
x = 2 and yt -.== i, we find, by substitution in (2), 

^1 -a 

in which the subscripts denote the given values of x and y. 

67. The expression for the derivative may sometimes be 
simplified by means of the given equation. Thus, if we have 

x yn —y*« j (!) 

or y n log x = x u logy, (2) 

on differentiating, we obtain 



dx 4- 7iy n ~ x log x dy = — dy^-n x n ~ x log y dx. 
x y 

Whence # = yiy-ux'-lozy) 

dx x {x n — ny n log x) 

and, by substituting from equation (2), 

^y ^ yn+z j — nlogx 
dx ~ x n + I 1 — n logjj'* 



(3) 



In this example, equation (1) is evidently satisfied by 
x=y — a for all values of <?, and equation (3) gives, except 

Avhen a — e n , 

dy 



dx 



— I. 



When a~e n , this derivative takes the form-, and its value 

o 

can be determined by the method given in Art. 116. 



§ IX.] EXAMPLES. J I 

Examples IX. 

1. U = xy e* + 2 y. <& = e* + ^ [y (i + x) dx + .r (i -f 2y) ^]. 

, x , y dx — xdy 

2. // = log tan—. du = 2 - - . 

y 2 • x 

y-sin2- 

3. u = log tan- 1 — . du = — ^— '— 2- — . 

y (^+^)tan-^ 

= ^ + ^ 

x +y 

du = [y~*- 2 VJxy)] yydx + [x - y - 2 4/^)] ^ d y 

2 ) /(xy)(x+yy 

e x y , ye x dx xixdy—ydx)^ 

t*- 2 +J 2 )* (-r 2 +^ (x 2 +j 2 )* 

c 4. — 1 - T — y 7 y dx — x dy 

6. u = tan ^-. ^/ = £ — -I-. 

x + y x 2 + y 2 

7 . u = 4/^4. du = "-Tlrf*-*^) . 

& x - V (-r 2 -J 2 )' dU ~ y V {x 2 - y 2 ) 

9. Given x = r cos 6, and y = r sin ; eliminate 6 and find dr ; also 
eliminate r and find dO. 



7 x dx + ydy , 
dr = — - — =^1 and 



dO 



dy — y dx 



i/(x 2 + y°) x 2 4 y> 

io. Given .r 2 (_y — i) + y % {x 4- i) = I ; find an expression in terms of 
x and j for—, and also its numerical values when j = 2. 



Deduce the corresponding values of x from the original equation. 

$-1 =-6,and# 

dx_]-i, 2 dx 



_6_ 
23 



72 



DIFFERENTIA TION. 



[Ex. IX. 



ii. Given xy 2 + x 2 y — 2 = o; determine the value of -j- by the 

method exemplified in Art. 66, also by making y an explicit function of 
x\ and find the values of the derivative corresponding to x = i. 



dy _ y + 2xy 
dx ~ 2xy -f x' 



x*- 4 



ix \Z(x* + 8x)' 



dy^ 
dx 



— — i, and 



dy_ 
dx 



= o. 



12. tan 



, x — a y — a 

1 tan - 1 ^— — = £. 



.r + # 



/ + a 



13. y = 1 + xev. 

14. (> — y)y n =x +y. 

15. (x 2 + j 2 ) 2 = a2 x 2 — Py 2 - 

16. ys n y= ax m . 

17. y 3 — sy sin ~ x x + x 3 = o. 

18. _y sin;zjtr — d:e n *+s' = o. 

19. j tan -1 .r — y 2 + jf 2 = o. 

20. j = tan(x +j). 



^ __ y + a- 

dx x 1 + ct 
dy £ y 



dx 2 — y 

dy — 2y 2 

dx ~ n {x^ — y' 1 ) — ixy 

dy _ [a- — 2 (x 2 + _y 2 )] .r 
rfr ~ [^"+"2 (^ 2 T7)]J* 

^ _ my 

dx x{i + #y)' 

</y y — .r 2 (l — ,r 2 )* 

- = 3/-^ 

(2y 3 - 



= 31/- 



^r 1 — y 



(1 — cot//;tr). 



dy _ y(y + 2x + 2x 3 ) 
~dx ~ (FT x 2 ) (/ + x*)' 



dy_ 

dx 



1 + J 2 



Miscellaneous Examples. 



1. y = 



*/(i + •*") 



^K X + 2 

dx ~ 2(1 + *)* 



§ IX.] MISCELLANEOUS EXAMPLES. 73 

2. y = a/ ' 1 ' 2 ~ - 1 " ' ' *Z- = (a n --t>"-)x 

V F-X 2 ' dx (tf*_^»)*(^ — ^)l' 

_ yfo + -y) i ^ _ y<i ( v'a- — j/ a) 

J ' V« + \/x' dx 2 \/.v */(a + x) ( 4/« + \/x) 2 ' 

A- y = ( V x - 2 V a ) Vi V a + V-*)- 4^ = : — • 

_ (-i- — i) (e x + i) £ J dy _ e x (.r f 2j — 2Jt' e x + 2t J — x) 

y ' e* — I "Zr ~ (P — \f 

6 = jrr+ . < 4£ = ,r^ x (.r? + x -' + logx+ I ) ^ 

dlr I — ,i-y + x log^r 

7. y = .**" _Z. = (n \ogx + I) j^'j"- 1 . 

dx 



\. y = ;r* 



J^- = .r** x* [ (log.r) 2 + log.r + - 
dx x 



. (1 + x-f , w . ^ 

9. j = log- L_ + £ tan- 1 .*-. 



(1 + xf dx (1 + x) (1 + .r 2 ) 

<^K _ x 2 
dx ~ 1 — x* 



10. y = logj I + M 4 — * tan-Lr. 

11. j = log[.r + ^/(.r 2 -« 2 )] + sec- 1 - ^-^Lj( J L±A\ 

a dx x r \x — a J 

2 sin -1 x . 1 — x dy 2.rsin — x x 

12. y = u log . — = . 

J 4/(1 — X) & 1 + x dx (j _ ^«)5 

i 3 . y = ( ' + y + 3**)* > ^ = _ (1 + -*•)' i 

•*" <** .r 2 (i + 3-^ + 3^ 2 ) § ' 



i a + 4/ (a 2 — x 2 ) . „ „. 

I4. J = tf log i ? _ |/(«- _ .r-). 



<fv _ |/(7z 2 — x 2 ) 

dx x 



74 DIFFERENTIATION. [Ex. IX. 



15. y = (1 - x-ysin-'x 



dy 1 — x 2 1 + 2x* „ 2N . , 

-f- = = — V(i — x 2 ) sin- 1 .*-. 

ax x x 2 



g- , /i — cos^r dy 

r 1 + cos^r dx 



17. j = tan- 1 a/ - • tan - . -/- = / ^ 

[_r a + £ 2 J dLr 2{a + <£ cos^r) 

, 1 dy 2 

18. j = sec -1 — . 



19. y = cos -1 



ix 2 — I <£r 4/(1— .r'-) 

.r 2 "— 1 </j/ _ 2nx n ~ 1 

x 2n + 1 * "Zr ~ ;tr 2w + 1 ' 



20. y = a cos -1 \/\b 2 — (a — x)*]. 



/ 1 — x 
21. jK = COS"T-2 i/ 

r 1 + x 



dy x 

~dx ~ \/[b- — (a — x)' 2 ] 

dy 4/(1 — x) 



dx 



(1 + xf 



rysy At — — , __ pCt tan X 

4/(1 + x 2 ) " dx ~(! + ^)f " 

£/j<? logarithmic differentials. 

(SxY — a8ox + 288 , 12 , . -N ^ 5^r 8 
23. y = !2_J _: ^ log (4 — $x). -4~ = ■ 

5 y 125 ( 4 -$xy ^ 125 &V4 > } dx ( S x- 4 y 



, x tan a- <^y « 2 tan a 

24. j/ = sin -1 



^ (^ 2 - jr 2 ) dx a 2 — x 2 4/ 2 — x 2 sec 2 a) 

25. j/ _ cos j/ (^ 2 _ ^ 2 J- ^ - ^ _ x2) v(a *_ ri) • 



§ IX.] MISCELLANEOUS EXAMPLES. 75 



26. y= tan-' ^ + '** * ^"^ 



t »/' — a) dx 2(1+ .n v la -j- bx) 



- . x cos.r 

27. y = log tan 

2 sm'jr 



, 1 4 x t/2 4 a- . _r 4/2 

v = log— Z v — x — + 2 tan- 1 1 — 

1 — x j^2 + xr 1 — jr 



1 / 1 125 , o;.i- 35.1- ,\ 

29. / = ; f - + — i + -i- 4 ^— + 5.V 3 ) 

(I 4-n 4 v .r 12 3 2 y 



1 I + x . , 1 + .r 4- .r- . . . x 1/3 

30. ^ = iog-Z^_ + 1 log — — ; . + 4/3 tan ~ 





dy 
dx 


2 
~~ sin\r 




dx 


44/2 

1 + a* 


+ 5 


1 - 1 
lop' 




10& 1 + x 


dy 




1 


dx 


x- 


(i+ar) 6 



I — X " I — x + x- 1 — x- 



dy 



dx 1 - x 6 

m 

31. _y = (i+ .v 2 / sin(w tan- x .r). 

tf 7 )' m — I 

-Z- = ;« (1 + x 2 ) -2 " cos [(;« — 1) tan -1 x\ 
dx 



. f.r |- 1 r . 2.r 4 I tfV 6x 

32. j = log— - — ' 2 j 3 tan - 



.r- + .r + 1 4/3 dx x 



33. Given u — x z + « sin z 4- a z cosz, and .r = a — a cosz ; prove 
that 

du /2a — x\i 

'dx' 



_P=^ 



CHAPTER IV. 

Successive Differentiation. 



X. 

Velocity and Acceleration, 

68. If the variable quantity x represent the distance of a 
point, moving in a straight line, from a fixed origin taken on the 
line, the rate of x will represent the velocity of the point. 

Denoting this velocity by v x we have, in accordance with the 
definition given in Art. 17, 

dx , . 

v * = w w 

In this expression the arbitrary interval of time dt is re- 
garded as constant, while dx, and consequently v x , are in gen- 
eral variable. Differentiating equation (1) we have, since dt 
is constant, 

T d(dx) 

The differential of dx, denoted above by d(dx) } is called the 
second differential of x ; it is usually written in the abbreviated 
form d*x, and read " d-second jr." The rate of v x is therefore 
expressed thus : — 

dv x _d*x 

W~(dty m 



§ X.] VELOCITY AND ACCELERATION. *]J 

The rate of the velocity of a point is called its acceleration, 
and is usually denoted by a ; hence we write 



«. = ££=;£, ( 2 ) 

dt at 



the marks of parenthesis being usually omitted in the denomi- 
nator of this expression. 

69. When the space x described by a moving point is a 
given function of the time /, the derivative of this function is, 
by equation (i), an expression for the velocity in terms of /. 
The derivative of the latter expression, which is called the 
second derivative of x, is therefore, by equation (2), an expres- 
sion for the acceleration in terms of /. 

A positive value of the acceleration a indicates an algebraic 
increase of the velocity v, whether the latter be positive or 
negative ; and, on the other hand, a negative value of a indi- 
cates an algebraic decrease of the velocity. 

70. As an illustration, let x denote the space which a body 
falling freely describes in the time t. A well-known mechanical 
formula gives 

* = \gf (0 

dx 
Hence we derive v x ———gt, (2) 

, dv x d^x f v 

and a *=-w=w= g - (3) 

In this case, therefore, the acceleration is constant and posi- 
tive, and accordingly v x , which is likewise positive, is numeri- 
cally increasing. 

71. When the velocity is given in terms of x, the acceleration 
can readily be expressed in terms of the same variable, as in 
the following example. 



78 SUCCESSIVE DIFFERENTIATION. [Art. J I. 

Given v x — 2 sin x ; 

, dv x dx 

whence -=— = 2 cos^r-^— ; 

that is, ol x = 2 cos x. v x = 4 cos ;tr sin ;r = 2 sin 2;r. 

The general expression for or*, when z^ is given in terms 
of x, is 

_ a^ _ dv x dx _ ^. r _ 1 d{y x ) , v 

*# d&ir <// x dx 2 dx 



Component Velocities and Accelerations. 

72. When the motion of a point is not rectilinear but is 
nevertheless confined to a plane, its position is referred to co- 
ordinate axes ; the coordinates, x and y, are evidently functions 

dx dy 

of /, and the derivatives — - and -f- , which denote the rates 

dt dt 

of these variables, are called the component or resolved velocities 
in the directions of the axes. Denoting these component veloci- 
ties by z^and v yi we have 

dx . dy 

v x = -g, and «,=-£. 



Again, denoting by s the actual space described, as measured 

from some fixed point of the path, s will likewise be a function 

ds 
of /, and the derivative — will denote the actual velocity of 

dt 

the point. (Compare Art. 48.) Now, the axes being rectangu- 
lar, and ^ denoting the inclination of the direction of the mo- 
tion to the axis of x, we have 

dx = ds cos <f>, and dy — ds sin (f>. 
TT dx ds . , dy ds . , 

Hence ' St = dt cos *« and * = * sm ^ ; 



§ X.] COMPONENT VELOCITIES AND ACCELERATIONS. 79 

or v x — v cos (f>, and v y = v sin </>. 

Squaring and adding, 

vl + «{ = ^ a . 

The last equation enables us to determine from the component 
velocities the actual velocity in the curve. 

73. If we represent the accelerations of the resolved mo- 
tions in the directions of the axes by a x and a yi we shall have, 
by Art. 68, 

*.= — and a y = J. 

These accelerations, a x and a yy will be positive when the re- 
solved motions are accelerated in the positive directions of the 
corresponding axes ; that is, when they increase a positive re- 
solved velocity, or numerically decrease a negative resolved 
velocity. 

Examples X. 

i. The space in feet described in the time t by a point moving in 
a straight line is expressed by the formula 

x = 4 St- i6f; 

find the acceleration, and the velocity at the end of 2\ seconds ; also 
find the value of t for which v — o. 

a = — 32 ; v — o, when / = i\. 

2. If the space described in / seconds be expressed by the formula 

4 



x = 10 log 



4 + / ; 



find the velocity and acceleration at the end of 1 second, and at the 
end of 16 seconds. When / = 1, v = — 2 and a = f . 



80 SUCCESSIVE DIFFERENTIATION. [Ex. X. 

3. If a point moves in a fixed path so that 

s= Vt, 

show that the acceleration is negative and proportional to the cube of 
the velocity. Find the value of the acceleration at the end of one 
second, and at the end of nine seconds. — J, and — -^-J-g. 

4. If a point move in a straight line so that 

• x = a cos \7tt, 
show that a = — %7t 2 x. 

5. If x = a e* + b €~* 9 
prove that a = x. 

6. If a point referred to rectangular coordinate axes move so that 

x = a cos / + b and y = a sin / + c, 

show that its velocity will be uniform. Find the equation of the path 
described. 

Eliminate t from the given equations. 

7. A projectile moves in the parabola whose equation is 

y — x tan a —f — — x\ 

2V cos a 

(the axis of y being vertical) with a uniform horizontal velocity 

v x — V cos a ; 

find the velocity in the curve, and the vertical acceleration. 

v — V{ V" — 2gy), and a y = — g. 

8. A point moves in the curve, whose equation is 

X s + y s — a*, 



§ X.] /-..WIMPLES. 8 1 

so that v x is constant and equal to k ; find the acceleration in the di- 
rection of the axis of y. a i ^ 

3X 3 y* 
9. If a point move so that v — V(2gx); determine the acceleration. 
Use equation (1), Art. 71. a = g. 

10. If a point move so that we have 

if = c — ttlog X, 
determine the acceleration. a = . 



2X 



n. If a point move so that we have 

2/* 



determine the acceleration. a = r. 

{x* + ?)* 

12. The velocity of a point is inversely proportional to the square 

of its distance from a fixed point of the straight line in which it moves, 

the velocity being 2 feet per second when the distance is six inches ; 

determine the acceleration at a given distance s from the fixed point. 

1 „ 
1 feet. 

2S 

13. The velocity of a point moving in a straight line is m times its 
distance from a fixed point at the perpendicular distance a from the 
straight line ; determine the acceleration at the distance x from the 
foot of the perpendicular. ol = m*x. 



14. The relation between x and / being expressed by 

/ a/— — V(ax — x*) — \a vers -1 — ; 
y a a 



find the acceleration in terms of x. oc = 5 

x 

15. A point moves in the hyperbola 

. / =/ x > + f 

in such a manner that v x has the constant value c ; prove that 



82 SUCCESSIVE DIFFERENTIATION. [Ex. X. 

and thence derive a y by equation (i), Art. 71. 

.a 2 ^ ~ 2 



16. A point describes the conic section 

y" — 2771 X + 71 X~ , 

v x having the constant value c ; determine the value of a • 



y 



Express vl 171 tewns of y, a,7id proceed as in Example 15. 



2 2 
771 e 

a y = r 



XL 

Successive Derivatives. 

74. The derivative of f(x) is another function of x, which 
we have denoted by f\x) ; if we take the derivative of the 
latter, we obtain still another function of x, which is called the 
second derivative of the original function f(x), and is denoted 
by f"(x). Thus if 

f(x) = x\ /'(*)=■&, and f"{x) = 6x. 

Similarly the derivative of fix) is denoted by f'"(x), and 
is called the third derivative of f(x) ; etc. When one of these 
successive derivatives has a constant value, the next and all 
succeeding derivatives evidently vanish. Thus, in the above 
example, f'"(x) = 6, consequently, in this case, f [v (x) and all 
higher derivatives vanish. 

The Geometrical Meaning of the Second Derivative. 

75. If the curve whose equation is 

y=f{x) 



§xi.] 



THE SECO.VD DERIVATIVE. 



83 



be constructed, we have seen (Art. 26) that 
g = /'(*) = tan *, 

(f> being the inclination of the curve to the axis of x; hence 

If now the value of this derivative be positive, 
tan <t> will be an increasing function of x, as in 
Fig. 6, in which, as we proceed toward the 
right, tan <f> (at first negative) increases alge- 
braically throughout. In this case, therefore, 
the curve appears concave when viewed from 
above. On the other hand, if f"(x) be negative, tan <f> will be 
a decreasing function of x, as in Fig. 7, in 
which, as we proceed toward the right, tan (f> 
decreases algebraically throughout, the curve 
appearing convex when viewed from above. 



Fig. 6. 



76. A point which separates a concave from 
a convex portion of a curve is called a point of Fig. 7. 
inflexion, or a point of contrary flexure. 

It is obvious from the preceding article that, at a point of 
inflexion, like P in Fig. 8, f"(x) must change 
sign; hence at such a point, the value of this 
derivative must become either zero or infinity. 

77. When a curve is described by a moving 
point, the character of the curvature is depen- 
dent upon the component accelerations of the Fig. 8. 
motion. For, if we put 

v x = c, or dx = c dt, 
c denoting a constant, we have 



/'{*) = 



dy 
cdt' 1 



84 SUCCESSIVE DIFFERENTIATION. [Art. yy. 

and hence ^ =£.*£= £k. 

Whence it follows that, if v x is constant, a y and f"{x) have 
the same sign, and consequently that a portion of a curve 
which is concave when viewed from above is one in which a y is 
positive when ol x is zero. 

Successive Differentials. 

78. The successive differentials of a function of x involve 
the successive differentials of x ; thus, if 

y = x\ 
we have dy — $x*dx, 

d*y = 6x(dxf + y?d*x, 
and d 3 y = 6{dx) 3 + iSxdxd 2 x + $x 2 d 3 x. 

In general, if 

y=f(*i 

dy =f'(x)dx, 

d'y ='■/%) {dxf + f\x) d*x, 
and d'y =f"\x) [dx) % + if"(x)dxd' l x +f'(x)d 3 x. 

Eqtiicrescent Variables. 

79. A variable is said to be equicrescent when its rate is con- 

dx 
stant ; since dt in the expression -— is assumed to be constant, 

dt 

dx is also constant, when x is equicrescent. 

In expressing the differentials of a function, it is admissible 



g XL] EQUICRESCENT VARIABLES. 85 

to assume the independent variable to be equicrescent, since 
the differential of this variable is arbitrary. This hypothesis 
greatly simplifies the expressions for the second and higher dif- 
ferentials of functions of x, inasmuch as it is evidently equiva- 
lent to making all differentials of x higher than the first vanish. 
Thus, in the general expressions for d 2 y and d 3 y given in the 
preceding article, all the terms except the first disappear, and 
it is easy to see that, in general, we shall have 

d"y=f"(x){dx)\ 
when x is equicrescent. 

80. From the above equation we derive 

dx n J x ' 

The expression in the first member of this equation is the usual 
symbol for the nth derivative of j/ regarded as a function of x. 
The 71th. differential which occurs in this symbol is always un- 
derstood to denote the value which this differential assumes 
when the variable indicated in the denominator is equicrescent. 

The symbol — - is frequently used to denote the operation 
dx 

of taking the derivative with reference to x, and similarly the 

/ d\ n d n 

symbol ( — j , or - — , is used to denote the operation of tak- 
ing the derivative with respect to x, n times in succession. 

Implicit Functions. 

81. Whenjj/ is given as an implicit function of x, the higher 
derivatives, like the first derivative (Art. 66), can in general be 
found only in terms of x and y ; hence the numerical values of 
these derivatives can be determined only for known simulta- 



86 SUCCESSIVE DIFFERENTIATION. [Art. 8l. 

neous values of x and y. The following examples will serve to 
illustrate the method of rinding such derivatives. 

Given \og(x + y) = x — y; . . . . . . (i) 

we obtain, by differentiating and reducing, 

(x +y -f i)dy + (i — x— y)dx = o; . . . (2) 

whence -4- = — . ....... (3) 

dx x+y+i K0J 

Differentiating and dividing by dx, 

dx 2 (x+y+jy 

dv 
substituting the value of -^-.we obtain 

dx 



<ty = 4(-^ + y) 
dx- (x +y+ i) ; 



(4) 



In like manner, the third derivative may be found. 

Simultaneous values of x and y are readily found in this 
case. Thus, if we put x + y = 1, we have x — y — o, whence 
x — \ and y = \ ; by substituting these values in (3) and (4) we 
obtain 

dy~ 

dx. 



= o, and 



dxU^L 



2- 



Differential Equations. 

82. An equation of which each term contains a first differ- 
ential is called a differential equation of the first order. Thus, 
equation (2) of the preceding article is a differential equation 



§ XL] DIFFER EX TIAL EQUATIONS, Sj 

of the first order between x and y. Such an equation is obvi- 
ously equivalent to a relation between x,y, and the first deriva- 
tive of y with respect to x (or of x with respect to y). 
By differentiating equation (2), Art. 81, we obtain 

(1 4- x + y) dy + (1 - * - j') <T.r + (<fy) J - W = °- • • (5) 

It is obvious that each term of such an equation must contain 
either a second differential or the product of two first differen- 
tials ; an equation of this character is called a differential cqua- 
tion of the second order. 

If x and y are functions of a third variable /, equation (5) 
may be converted into a relation between x,y, and their deriva- 
tives with respect to /, by dividing each term by (dtf. When 
one of the variables (x or y) is regarded as a function of the 
other, the independent variable is made equicrescent. Thus, 
if x be regarded as the independent variable, equation (5) be- 
comes 

(1 + x +y) d*y + (dyY - {dxf = o, 

(i + .^)g + (g)^i=o, 

a relation between x, y, and the derivatives of y with reference 
to x. 

83. A given equation and the differential equations derived 
from it are sometimes so combined as to produce a differential 
equation having certain desired characteristics. Thus, if it be 
required to derive from 

y = sin -1 ^: (1) 

a differential equation free from transcendental functions and 
radicals, we have 

d_y = J . 

d.x v'(l — -i' 8 ) ' . 



88 SUCCESSIVE DIFFERENTIA TION. [Art. 83. 

. d*y x x dy 

whence -y4 2 == Tx — - 2 • -f- , 

ax 1 (1 — x *\t 1 — x 2 ax 

, 2N dy dy 

a differential equation of the required form. 



Examples XI. 

1. Find the second derivative of sec x, and distinguish the concave 
from the convex portions of the curve y = sec x. Also show that the 
curve y = log x is everywhere convex. 

2. Find the points of inflexion in the curve jy = sin x. 

3. Find the point of inflexion of the curve 

y = 2X 3 — 3JC 2 — 1 2.x + 6. 

The point is (£, — J). 

4. Show that the curve y = tan x is concave when y is positive, and 
convex whenj is negative. 

5. Find the points of inflexion of the curve 

y = x* — 2x z — 12JC 2 +■ iijc + 24. 

The points are (2, — 2) and (— 1, 4). 

6. Uf(x)= i±|, find /*(*). / V W = ^~ . 

7. If/(*) =-, find/ ^). / (*) = - -* ^ '—. 

8. If jy is a function of x of the form 

A x n + Bx n ~ x + • • • + Mx + iV, 

prove that , -~ = 1. 2. 3 • • * n A. 



§ XL] EXAMPLES. 89 

9. If/ (*) = b", find/ v (x). f x (x) = a" (log bY b ax . 

10. If/ (*) = x s log (in*), find/ IV (*). r (*) = |. 

11. If/ (*) = log sin *, find/"' (*). /'" (*) = ^^. 

12. If/ (x) = sec jc, find/" (#) and/'" (#). 

/" (#) = 2 sec 3 jt: — sec x, and/"' (#) = sec jc tan x (6 sec 2 x — 1). 

13. If/ (#) = tan x, find/'" (#) and/ ,v (#) 

/'" (x) = 6 sec 4 * — 4 sec 2 x, and/ IV (*) == 8 tan x sec 2 .* (3 sec 2 .* — 1). 

14. If/ (*) = x r , find/" (*). f" (x) =x*(i + log xf + jc 1 - 1 . 

i 5 . if j, = e\ find ^. S = - A-(i + 6* +6x-) e«. 

16. If j- = <T* 2 , find^ 3 . ^ = AX (3 " 2^ 2 ) £- x \ 

17. IO = log (**+*-*>, ^d g. 



- 8 



^C 3 (£*+£-*) 3 ' 



o Tf z c a d y a d y 

18. If y = — , find -7-5 and -^— 4 . 

^ £ T — 1 dx 1 dx 



d'y _ e°- x + e x d'y _ e x + iie 9 * + n£ 3x -f £ 4x 

dx 1 ~ (F^y ' an ^ ~ T fX -i) 5 



19. If y = sin ' #, find -j- A 



d'y d*y _ gx + 6-v : 



</* 4 ' dx* ( ! _ ^)£ 



73 
20. If y = £ sln *, find -4'. 



-j^ = — £ sini cos a: sin jc (sin x + 3). 



90 SUCCESSIVE DIFFERENTIATION. [Ex. XI. 

21. If 7 = — : • , find -4 . -4- :- — — — *—.. 

I + log JP dOT /MP 37 ( I + log X) 

22. Find the value of d 3 (s x ), when ^ is not equicrescent. 

d 3 (s x ) = a*(dxf + $s*d*xdx + e x d*x. 

d 3 

23. Find the value of -rg- (sin e), being a function of /. 

d 3 , . x /^\ 3 . <# </ 2 </ 3 

_( sme ) = _cos^-|-3Sin 6 -.-^ + cos e -^- 3 . 

24. If jy is a function of x, and if u — y* logy ; find jy . 

_ = ( 2 log^ + 3)^_j +J ( 2 lo gj , + i)^. 

x d u 

25. If & = — , y being a function of #, find -=-$ . 

*/ 2 z/ _ 2 dy 2x/ dy\* x d*y 
dx* f dx y 6 \dx J y 1 dx* ' 

27. If _y = tan (# + j>), find -7^. 

28. If f + y = x\ find ^ 3 . 

</ 2 y 

29. Given £ x + jc = e y + y, to find -^ 



30. Given e 2 ' + xy — £ = o, to find -j- 2 



*/> 


2(5 + 8/4-3/) 


d& 3 


y 




^ 3 j 24^ 




dx* (1 -f 2jy) 5 


rt^ 


_(6^-l)(*-j) 


^ 2 " 


(* + l) S 


^. 


(2 —y)e 9 + 2X 


dx" ' 


~ } (** + *)■ 



§ XL] EXAMPLES. 91 

31. Given y — 3^aj' -h a 3 = o, to find — - a . 

d'y _ 2# 3 .%t 

.7? ~ ~ (/ - **)* 

32. Given xy = a e x + ^£~ x , to deduce the following differential 

equation : — 

d'y dy 

x-^— + 2 — xy = o. 

dx dx 

$$. Given y = 6 x sin.v, to derive 

dy dy 

dx' 2 dx 

34. Given y = a sin x + b cos x, to derive 

dy 

35. Given y = a cos log x + b sin log x, to derive 

„ dy dy 

* 2 -t4, 4- x-f- +y — o. 

dx' dx J 

36. Given y — (sin -1 x)\ to derive 

, 2 . d 2 y dy 

(1 — X-) -f\ — x-f- = 2. 

37. Given y = a e x + b s~ x + <r sin (x + w), to derive 

dy 

dx~>- y=0 ' 

£ x 4- €~ x 

38. Given y = — , to derive 

° J 8 X — £ _x ' 

Jx = '- y ' 

39. Given y — log [x + ^/(a 2 + x 2 )], to derive 



92 SUCCESSIVE DIFFERENTIATION. [Ex. XI. 

40. Given y = [x + <y/{x 2 + i)]"\ to derive 

(x* + 1) ~ + ■* -r- — n?y — o. 

41. Given the equation of the circle 

(* - «) 2 + (y - bf = c\ 
to derive a differential equation independent of a and b. 



dx 1 



■ + ay 



XII. 

Expressions for the nth Derivative, 

84. The expression for the /zth derivative is readily found 
in the case of certain functions. Thus, if 

f(x) = \ogx, 

/'(*) = .-*-', 

/>"(*)= 1-2 X-\ 

and / IV (*) = — 1-2-3 *- 4 ; 

whence, it is evident that 

f n {x) = (—i) n - l i.2--'{n—\)x- n (1) 

Again, if / (x) = sin x, 



>i XII.] EXPRESSIONS FOR THE N Til DERIVATIVE. 93 

/' (x) = cos # = sin (;tr + £71), 
/" (x) = sin (* + 7z), 
/" (.r) = sin {x + | *), 

therefore /" (V) = sin ( ;tr H ?rj (2) 

85. Artifices of a less obvious character are sometimes neces- 
sary in expressing the ;zth derivatives of functions. Let 

y — e ax cos {bx) y (i) 

then dy = e ax [a cos {bx) — b sin (bx)] dx. 

Employing an auxiliary constant a determined by 

b — a tan a, (2) 

we have 

-2- = e ax [cos (bx) cos a — sin (bx) sin a~\, 

dx cos a 

= e ax cos (bx + «)• 

cos a 

In like manner, we obtain 

d'v d 2 , , s 

—4 = — * ax cos (bx 4 2a), 

dx~ cos'' a 

d n v d l 

and in general -^~ = € ax cos (for 4 # a), 

s dx n cos 71 ^ v J 

or, since, by equation (2), 

COS tf = f— TTT-, 

— 4 = (V 4- £ 2 ) 2 ^cos (for 4- n tan- 1 -^ . . (3) 



94 SUCCESSIVE DIFFERENTIATION. [Art. 86. 

86. To find the nth derivative of the function 

* = ^h' (I) 

we put x — a cot 0, (2) 

being an auxiliary variable, whence 

y = — sin 2 0, 
a* 

1 dy 1 . „ n dd , . 

and -f- ■- — 2 sin (9 cos -=- 13) 

dx a 2 dx w ; 

Differentiating equation (2), we have 

dx = — a cosec 2 0^0, 

dd sin 2 
or -j- = ; 

substituting in equation (3), 



% = - -L sin 20 sin 2 0, 



whence 



-■=4 = r ■ 2 sin (sin 20 cos + cos 20 sin 0) — , 

dx 2 a 3 v V* 

'— sin 30sin 3 0. 



In a similar manner, we obtain 

d % y 1-2-3 • n • 4/j 
— 4 = ^ sin40sin 4 0, 

and in general 

g = '(_ I)» LU__l« s ; n [( „ + r) ^ (s ; n ^ + 1 . . , (4) 



§ XII.] LEIBNITZ' THEOREM. 95 

Since we have, from equation (2), 

. a 

sin 6 = 



V(a 2 + ;tr 2 ) ' 
equation (4) may be written thus — 

;/-(- i)"~ ^-^sin ["(«+ 1) tan" 1 -]. . .(5) 
a\ar-\-x") - *- - 1 

O-T C d 4- -\ X a 

87. bince ——tan — = — - 



d n x d"~ l 

we have -=- tan l - = a — . , 

«j; n tf ^mt" l \a~ 4- .r 



Hence we obtain from equation (5) 

- (n - n . r 

« tan' 



^ *. 1 •*" / \« 1 I,2 -3 • • • ( n — • 
-=- tan -1 - = (— i)"- 1 - v ; sin 

dx n a f 2 . '- 

(a 2 + •* ) 2 



Leibnitz Theorem. 

88. We proceed to deduce an expression for the ;zth differ- 
ential of the product of two variables in terms of the successive 
derivatives of these variables. 

From d(uv) — udv -j- vdu, 

we derive, by successive differentiation, 

d\uv) — u d*v + 2du dv 4- v d*u, 

and d\uv) = ud*v 4- $dud*v + id*ndv + vd % u ; 

from which the law of the indices in the expansion of d n 'uv) is 
readily inferred. We also see that, when n = 2 and when n — 3, 
the coefficients are identical with those in the expansion of 



g6 successive differentiation. [Art. 88. 

(a + b) n . That this is true for all integral values of n may be 
shown in the following manner. Assuming that 

d\uv) — ud n v + n du d' l ~ 1 v + n ^ n ~ 1 f dhi dv n ~ 2 + • • • , . . (i) 

we have, by differentiating, 

d n + \uv) = ud n + 1 v-\-(n+i)dud n v-\-n['i +?—± \d'ud n ~ l v 

+ n 7 L^±\ l + 1 lZLl]d 3 ud»- 2 v+ ..., 
2 L 3 J 

or, reducing, 

d n+l (itv)=iid n + l v + (n+\)dud n v+^- l ^-d'ud n - l v+ 

in which the coefficients follow the same law. Hence, if equa- 
tion (i) is true for any value of #, it is true for all greater values 
of ;/; now equation (i) is true when ».= I, therefore it is true 
universally. 

The result expressed in equation (i) is called from the name 
of its discoverer the Theorem of Leibnitz. 

When u and v are functions of the same independent varia- 
ble we may, by dividing by (dx) n , convert the differentials in 
equation (i) into derivatives. 

89. By means of this theorem the nth. derivative of the 
function 

x 



<? + x* 



may now be deduced, since the nth. derivative of — = has al- 

c? + x l 

ready been obtained in Art. 86. Putting u = x in equation (i), 

du 

-r- — I, and all higher derivatives of u vanish; therefore we 

, dHxv) d n v , d n ~ l v 

have -^ 1 ^ x ^ + n 'd^- 



I XII.] 



LEIBXITZ' THEOREM. 



97 



Putting v= ^— — = in this result, and employing the value of 
a- + x r y & 



d n 



dx n Vf + * 



-) given in equation (4), Art. 86, 



dx n \a' 2 + x- 



(" H 



2 • • • n 



a n+ 



sin [(« + i)0](sin 0) n+1 

- sin (» 0) (sin d) n 



d n ( x \ 1-2 • • • n , . „ 



and, since x = a cot (9, [equation (2) Art. 86] 

un[(* + i)#]cos# 

— sin (n 6) 
or, since sin (n 6) = sin [(« + 1) 6] cos (9 — sin 6 cos [(;z + 1) 6] 

(- o b I '';;; ; ' (^ *)« +i cos k» + o ^. 






90. An important result is obtained by applying Leibnitz' 
theorem to the function 

tie**, 



u denoting any function of x 
We obviously have 

d 



dx r 



(e ax ) — a n e ax ; 



whence putting v = £**in equation (1), Art. 



dkr n 



#£'■ 



rt-T 1 £rt-^' 



ax 1-2 d!r 2 






This result is frequently written in the symbolic form 

d n f \ ._/ . d\ n 
~dx n 



— E ax [ a + —J 2/, 



9 8 



SUCCESSIVE DIFFERENTIATION. 



[Art. 90. 



which is to be understood as indicating that ( a + -7- J is to be 

expanded as if it were an ordinary binomial, and that u is to 
be affixed to each term of the expansion. 



The Numerical Values of Successive Derivatives for 
Particular Values #f the Independent Variable. 

91. The theorem of Leibnitz is sometimes employed in de- 
ducing a general relation between two or more successive deriv- 
atives of a function, by means of which their numerical values 
may be computed for given values of x. Thus, if 



y = sin (msin~ 1 x), 



(1) 



we have 
and 



dy _ m cos \in sin x x\ 
~dx~ V(i - **) 



dy 

dx 2 



— iri* sin [m sin -1 ^] + m cos [m sin _1 ^r] 



V(!-^) 



(*) 



(3) 



Whence, (1 — x*) —^ — x -4- + f£y — o. 

dx* dx 

Taking the «th derivative of each term by means of Leibnitz, 
theorem, we find 



(1 — x 2 ) "- — ~ — 2n x — — -?- — n(n — 1) . 



d n+ *y 
dx n + T 



dx 1l + 



d n y 
dx r; 



% rl_ ft _ 

dx n + L dx n 



+m 



d n y 
dx' 1 



= o, 



§ XII.] NUMERICAL VALUES OF DERIVATIVES. 



99 



or 



//" -] d n + l v d n v 

0-^)^-(-+0-4^ + K-^)^ = o, . . .(4) 

the general relation required. 

92. Equation (4) may be employed to compute the numeri- 
cal values of the derivatives of y corresponding to x — o. Sub- 
stituting this value of x, equation (4) becomes 



d n+2 y 






Now from (2) and (3), we obtain 



dr\ 



d*y~\ 



- m, and "^ = o ; 

ax jo ax 2 Jo 



hence when 11 is even 



d n y- 



dx n _j o 
and, putting n equal to 1, 3, 5, etc., we have 



dy 

dx l 



d h y 



m(\ — in-), — - 5 I = ;«(i — nf) (9 — m 2 ), etc. 

&X _| 



Examples XII. 

1. y = x m ; find -^- . -7— = ndm — 1) ■ • • n — n + 1) x m ~ n . 
dx a dx n 



2.y 



(a - x)' 



d n y _ 



dx n 



3. y = cos vi x. 



= m(tn + 1) - ■ • n + n— 1) (a — x) m ". 
d"y „ / , n s 

— -^ = ;;/" COS (/#.* H 7t). 

ax 2 



IOO 



SUCCESSIVE DIFFERENTIATION. [Ex. XII. 



4- y 

5- y 



log 6 (a + x). 
log (i — mx). 



ay 
dx" 



<Fy _ , x..j i-2- • • (n- 1) 
<fc n V ; log£(a + #)" ' 



— i«2 • • • (;z — i)^ M (1 — mx) n , 



6. y = 



d n y 



£ ax sin bx. -p- — (a 2 + fi 2 )- £ ax sin \ fix + n tan" 



&v -f /ztan l — , 



• . A a . y m{m — 1) • • • 1 

j. y = x m log x, prove that W + I = — 7 



8. y = £ xcosa cos(^sin^). 



-^ — 6 -cosa CQS ^ gin a + na y 



g. y = cos x. 



<Ty n i / , « 

—^ = 2 71 " COS 2# + - 7T ) . 
ax \ 2 



10. y 



1 1 

# — x a + x 



d n y _ i-2 • • • n 



dx n 



a-xf +l ^ {a + *)" + 1 J' 



11. y = 



a 2 - x l 2 



a — x a + x 

d n y _ 1 • 2 • • • 11 
dx n ~ ' ~2 



r (-i) w i' 

Ja - x) n + l {a + x) n + 1 J 



12. y = x£' x ; find — , by Leibnitz' theorem. 



13. y — x^e x . 



dj 

dx 11 



g.= a -»(*+.*o* 



\ll(ll — i) -f 211 X + X 2 ]£ x . 



§ XII.] 



EXAMPLES. 



IOI 



14. y = x 3 z\ v being a function of x. 



d'y 



d"v 



7)1 - 1 

a v 



dx 



n = *" -T~ n + 3>** 2 T^n + 3»(« - 1) * 



</ n 'V 



^t: 



^ 



dx n ~ 2 
+ 72(72 — 1) (n — 2) 



15. j; = x *\ogx. 



<r-*v 



dx n - A 
When n > 3, 



d&" 



(-I)- 1 !* 



16. y = x sin x. 

*7- y = ( T ~~ x) n x n . 

d n y 

Tx^ 1 ' 2 " 



|_« — 1 n — 2 « — 3 J 
x 4- « — J — n cos f .# + n — J * 






• ;z i (1 — x) n — n 2 x (1 — x) n ~ l 



[ 



^(^ — 1)" 



1-2 



* 2 (l - *)' 



18. If y = * m £«*, prove that 



dfc* 



and thence show that 



+ ^ ^-^" 2 M (m - i)x m ~* + 

1-2 



■J 



dx'' 



t**x n — a' 






6 ax *" 



19. If ^ == sin *#, derive, by applying Leibnitz* theorem to the 
differential equation obtained from this function in Art. 83, the result, 



(*--0 £3 -<«+->.* 



dx n+i n 'dx n ~°' 



102 



SUCCESSIVE DIFFERENTIA TION. 



[Ex. XII. 



and thence deduce by the method of Art. 92 the values of the suc- 
cessive derivatives when x = o. 



dj 
dx 3 



— 1, 



and when n is even 



d -X 

dx\ 

dy~ 

dx n 



3> ^ 



= 3 2 -5 2 > etc. ; 



o. 



dy 
20. If y = tan -1 a:, we have (1 + x*) — 1 = 0. Hence derive, 

(IX 

by means of Leibnitz' theorem, 

. d n + 1 y d n y . ,d n ~ l y 

(I + x) *& + 2nx ^> + < n ~^i^ = °- 

and the values of the derivatives when x = o. 



dx 



and when n is even 



dx* 

d n y~ 

dx 11 



d_y 

dx" 



= 2-3-4: 



o. 



21. Given y — log [x + V{cr + x 2 )], derive 



d n+2 v d n+1 v d n v 

and thence (a- + .v 2 ) -^ + ( 2 « + 1)* ^ + * 2 -^ = o ; 



dx" 



dx n 



also the values of the derivatives when x — o. 



dx 



and when ;z is even 



dy 

dx\ 

dy 

dx 11 



d h y 



a* ' dx*_j Q 



= o. 



22. Given j = [x + ^(a 2 + ^ 2 )] M , derive 

{ j + s) d -£ + xf-My = o. 

v 7 dfo dx 



§ XII.] 



EXAMPLES. 



103 



and thence (,r + *•) ^ 2 + (*■ + r)*^ + (rf - „») g - o ; 



also deduce the values of the derivatives when x = o. 



dS*" 



dx 



= ma" 



d_2 

dx 



0o=^ Sl--**-*^ 






— ■wMwfl 



m\m* — 4)a m ~*. 



23. Given xy — as* + bs~% derive 



dy dy 

X dx* +2 !x- = Xy > 



and thence x--~ 2 + {11 + 2) .^-~ — x-~-—n —--A. 
d# "*" ax ' dx dx 



24. Given%j^ = a cos log a; + b sin log #, derive 
d' 2 y dy 



and thence 




dx? Vl _ 




also the values of the derivatives when x = 1. 


4" 


=4 S" 

1 dx _ 




= 3<* 

1 


+ *> and 31 



= o 



10(2. 



CHAPTER V. 

The Evaluation of Indeterminate Forms. 



XIII. 

Indeterminate or Illusory Forms, 

93. When a function is expressed in the form of a fraction 
each of whose terms is variable, it may happen that, for a cer- 
tain value of the independent variable, both terms reduce to 

zero. The function then takes the form - , and is said to be 

o 

indeterminate, since its value cannot be ascertained by the ordi- 
nary process of dividing the value of the numerator by that 
of the denominator. The function has, nevertheless, a value as 
determinate for this as for any other value of the independent 
variable. It is the object of this chapter to show that such defi- 
nite values exist, and to explain the methods by which they 
are determined. 

The term illusory form is often used as synonymous with 
indeterminate form, and these terms are applied indifferently, 

not only to the form - , but also to the forms — , oo- o, cq — oo, 
o oo 

and to certain others whose logarithms assume the form oo-o. 
When a function of x takes an illusory form for x—a, the cor- 
responding value of the function is sometimes called its limit- 
ing value as x approaches the value a. 

94. The values of functions which assume illusory forms may 



§ XIII.] INDETERMINATE OR ILLUSORY FORMS. 105 



sometimes be ascertained by making use of certain algebraic 
transformations. Thus, for example, the function 

a — V(a 2 — bx) 
x 

takes the form - when x = o. 
o 

Multiplying both terms by the complementary surd 

a + V(a* — bx), 

bx b 



we obtain 



x\a + V{a i - bx)] a + V(a* - bx) 



The last form is not illusory for the given value of x, since the 
factor which becomes zero has been removed from both terms 
of the fraction. The value of the fraction for x = o is evi- 
dently — . 
2a 

The following notation is used to indicate this and similar 
results ; viz., 



V(a 2 - bx)~\ _ b_ 
2a 



a 



the subscript denoting that value of the independent variable 
for which the function is evaluated. 



Evaluation by Differentiation. 

95. Let - represent a function in which both u and v are 
u 

functions of x, which vanish when x = a ; in other words, for 

this value of x, we have it = o, and v = O. 



106 EVALUATION OF INDETERMINATE FORMS. [Art. 95. 



Let P be a moving point of which the abscissa and ordinate 
are simultaneous values of u and v (x not 
being represented in the figure) ; then, de- 
noting the angle POU "by 0, and the inclina- 
tion of the motion of P to the axis of u by tj>, 
we have 




Fig. 9. 



tan (9 



and tan <j> 



dv 
du 



At the instant when x passes through the value a, u and v 
being zero by the hypothesis, P passes through the origin ; the 
corresponding value of 6 is evidently determined by the direc- 
tion in which P is moving at that instant, and is therefore equal 
to the value of (f> at that point. 

Hence the values of tan 6 and tan <j> corresponding to x = a 
are equal, or 



v~\ _ dv~\ 

l<<\x=a duj x= a 



therefore, to determine the value of - for x — a. we substitute 

u 

du 
for it the function -r- , whose value is the same as that of the 
du 

given function, when x — a. 

96. This result may also be expressed in the following man- 
ner : let f(x) and (f>(x) be two functions, such that f(a) — o, 
and (j){a) = o ; then 

As an illustration, let us take — - — . When^r— i,thisfunc- 

x — 1 



tion takes the form — ; by the above process, we have 



§ XIII.] 



E VAL UA TION B Y DIFFERENTIA TION. 



107 



x ~\ _ x 

tJi"'T 



log^r - ] _ x~ l 

X 



the required value. 

97. Since the substituted function — or • * } ) frequently 

an (p (x) 

takes the indeterminate form, several repetitions of the process 

are sometimes requisite before the value of the function can be 

ascertained. 

For example, the function — — takes the form - when 

v o 

8=0] employing the process for evaluating, we have 



1 — cos 8 



H _ si 
Jo" - : 



sin 6~~\ 
26 



which is likewise indeterminate ; but, by repeating the process, 
we obtain 



— cos 



sin 6~ 
~W J 



cos 8' 



2 Jo 



= h 



98. If the given function, or any of the substituted func- 
tions, contains a factor which does not take the indeterminate 
form, this factor may be evaluated at once, as in the following 
example. 

The function 

(1 — x}e x — 1 
tan 2 x 

is indeterminate for x = o. By employing the usual process 
once, we obtain 



, (i - X) £ x — i" 



tan'^r 



x& 



2 sec x tan x. 



which is likewise indeterminate ; but, before repeating the pro- 

e x 
cess, we may evaluate the factor — 

this factor is — J ; hence we write 



2 sec 



d 



The value of 



108 EVALUATION OF INDETERMINATE FORMS. [Art. 98. 



(I 


tan" x 


\~ 


xe* 


tdLtlX 




2 sec 2 
sec' 2 ^r_ 


.#■ tan xA° 
\ 

- 1 

2* 





99. When the given function can be decomposed into fac- 
tors each of which takes the indeterminate form, these factors 
may be evaluated separately. Thus, if the given function be 

(e* — 1) tan 2 :r 



may be employed. We have 

tan x~\ . s x — i~| 
= 1, and = I ; 

X — lo X lo 

hence the value of the given function is unity. 

When this method is used, if one of the factors is found 
to take the value zero while another is infinite, their product, 
being of the form o- 00, must be treated by the usual method, 
since o • 00 is itself an illusory form. 

100. Another mode of decomposing a given function is that 
of separating it into parts, and substituting the values of such 
parts as are found on evaluation to be finite. 

As an illustration, we take the expression, 



11 — . ~ __ 



'!■ 



Each of the fractions into which this function can be decom- 
posed being obviously infinite, we first apply the usual process, 
thus obtaining 



S XIII.] EVALUA TION B Y DIFFERED TIA TION. 



IO9 



_ 2p - r - g-*)(V +£-r) - 4r(V - + g--*) — 2;tr 2 Q r — £ ~ x ) 
Separating this expression into two fractions, thus, — 



1 



_ l " + it*) (e* - £-* — 2X)~ 

u — 

2X* 



2X 



the latter is found on evaluation to have a finite value, and the 
expression reduces to 



a* — e~ x — 2x 



Hence 



__ £f + e~* — 2~] _ £* — £~ x ' 

^x 1 Jo 6x 



-i = -§. 



Examples XIII. 



1. Prove 



sin x 



tana: 

h IT 



e * — 1 
= 1, and -r— — 

x 



1. 



These results are frequently useful in evaluating other functions. 
Evaluate the following functions : 





6 X — £~ x 




log(i+*)' 




a n - x n 


3- 


log a — log a ' 


4- 


x* — 5^+ 7a— 3 


a s - 0? — $x - 3 ' 


5- 


a* — 8JC 3 + 2 2JC 2 — 24^ + 9 


* 4 — 4JC 3 — 2X' + I 2JC + 9 


« 


are*"— * 2x - a: + 1 



when x = o. 



jc = #. 



^ = 3- 



# = 3- 



a: = o. 



2. 

1 

4' 

1 

4* 

— 1. 



HO EVALUATION OF INDETERMINATE FORMS. [Ex. XIII. 

sin x — cos x . „ . 

7. — , when x = ±tc, \\2. 

Sill 2X — COS 2* — I 



log* 



* V(i-x) ' 

X — If 



X = I. 



* = o. log - . 



4/(1 +Q(i-*) , c ■". 4/2 

10. — -^ -, (See Art. 98), x — 1. — 



II. 


« 2 -*v 


x> K c 


12. 


€ mx _ £ ma 


x — a ' 




a slQX - a 


J 3- 


log sin x ' 


14. 


1 — cos X 


#log(i + x) ' 


TC 


Vx tan x 



x\ I 

x = o. — 



9- 



2 



.£ = J7T. <z log tf. 



I 
x = o. — 

2 



X =0 . I. 



(*■- l) 3 

Put in the form 1/ • . See Art. 09 and Ex- 

J f e x —i x a x — 1 

atnple 1. 

4/* — j/a 4- V(x — a) _ 1 

l6 ' V(**-a*) ' *"* ~V~(2aY 

xVi^X — 2X*) — X 5 8l 

17. ^ / , x=i. —. 

I-X* 2 ° 

„ (a* + ax 4- * 2 ) T2 " — (a 1 — ax + jc 2 )* 

18.- , , x ) 71 -, *=°- VA 

(^ + *) 2 — (tf — jc) 2 

Multiply both terms by the two complementary surds. See Art. 94. 



§ XIII.] EXAMPLES. HI 

19. v ' whenjf = a. — ; 7. 

V (a 8 -* 8 )* + (*_*)*' i + aVz 

Divide both terms by (a — #)*. 



sin jc — x cos .r 



^U. 


jp — sin jc 


21. 


«* — f~* — 2X 


x — tan jc 


22. 


(x — 2)e x + x -\- 2 


*(*•-!)" 


*3- 


or" — # 


1 — # + log # ' 




tan x — sin # 



D . sin # 1 sec .* — i~| 
jrut in the form • , . 

X |o X |o 



27. 



28. 4 



X 

i 7T — tan ~ l x 



jf = o. 2. 



X = o. 



X = o. 



# = o. 



I 
6' 



# = 1. — 2. 



(*— i) 2 + sm 3 x" — 1)2 

(.* + i) (* - l)« 

1 — jc + log # 

26. -, 2-^, 3?= I. -I. 

1 — 4/(2.*: — xy 
smx — log (£*cos.#) 



sin (leg x) > 



X 


= 0. 


1 
2' 






1 


X — I. 


2(1— 11)' 


X = 0. 


(1 +# 2 ) sec 2 #. 



x" - e 

tan (a + x) — tan (# — #) 
tan -1 (a + x) — tan -1 (# — x) 

x sin # — h n , 

cos # 



112 EVALUATION OF INDETERMINATE FORMS. [Ex. XIII. 



~x ~sin x 



£~ — C 

31. ; — , when x = o. 1. 

.# — sin ^ 



32.— , x=i. \oga — 1. 

• log # 

# 3 — i 4- (.* — i) 2 3 

33. — -r 1 -, * = I - -"• 



" (^_ I )I__^ + I ' 



cos x (i — ^) 

34 - v(»*-«r * = °- 

x 2 — aV ax 

3S-^pr a . * = «■■ 3* 



. tan /z.# — /z tan x 

36. : : ■ X = Q. 



4/2 — cos x — sin x -, , , 

37. i = , ^=i^r. -tV2. 

log sin 2jc 



. * + * 3 - (2/2 + l)* 2n + 1 + (20 - l)^ 2 " + 3 

38. ^3-^ -^ , *=r. 



w* sin «# — s sin /«# 

39. — , x = o. 

tan «# — tan mx 



tan m — tan mx 1 

40. . , „ j-r-, a? = o. — -— . 

sin («"jp — ;» jp) 02 + n 

m x sin «# — n x sin a»# 

41. , //z = /z. 

tan /z# — tan mx 

n x ~ l {n cos nx— sin /zx) cos*nx. 

In solving this and the following example, x and n /way be regarded as 
constants, and m as a variable. 



§ XIII.] /..WIMPLES. 113 



tan nx — tan mx sec'"' nx 

42. . . m — n. . 

sin \tfx — mx) 2n 

43. Prove that, if/(o) = o, —r j = 1. 

" X ./\- X )-lo 

The given function may be put in the form 

x - f(x) ' 

ivhen x = o, the second factor is unity by Example 1. 



XIV. 



The Form — 

CO 



fix) 
101. Let v^ denote a function which assumes the form 
6 x l 



— when ;r = a, then we have 
00 



I 



(I) 



/(*) 



The second member of this equation takes the form — when 

o 

x — a\ we therefore have, by equation (1) Art. 96, 

_J_ fi (a) 

fia) _ 4*a) __ __[?U7 1 l_ Ka) j/(*)) 8 . ,. 

Ha) -_!_-_ fia) -f\a) \ fta) I ' " ' { 

whence, if ,, ; is neither zero nor infinity, we infer that 
4{a) 



114 EVALUATION OF INDETERMINATE FORMS. [Art. 101. 

f(a) = f(a) ,* 

Hfi) *'{*) w 

This formula, it will be observed, is identical with that employed 

when the function takes the form — . 

o 

102. In the above demonstration, it is assumed that both 

-j~ and -th-A have definite values ; and also that the re- 
<p(a) (p (a) 

quired value is neither zero nor infinity. It will be shown in 

the next article that the latter assumption is not essential to 

the existence of equation (3) ; but when the former assumption 

does not hold true, this equation ceases to be applicable. Thus, 

in the case of the function 



x -f- cos^ ? 



which takes the form — - , when x = 00, we have 

f'{x) __ I — COS^r 
(f>'(x) I — sin x ' 

An expression which has no definite value when x is infinite, 
since sin 00 and cos 00 have no definite values. The given 
function has, nevertheless, the definite value unity ; for 



sin x 
x — sin x* 



x -f- COS.*. 



cos x 
I +■ 



= I. 



00 



103. When the value of —A- is either zero or infinity, equa- 

<f)(a) 

* Cakul Diffdrentiel, par J. Bertrand, Paris, 1864, p. 476. 



§ XIV.] THE FORM -£. 115 

tion (2), Art. 101, will be satisfied independently of the exist- 
ence of equation (3) ; we are not justified therefore, when this 
is the case, in deriving the latter from the former. The follow- 
ing demonstration shows, however, that equation (3) holds in 
these cases also. 

First, when the value of - r ;{ is zero, by adding a finite 

<p(a) 

quantity 11 to the given function, we have 

/(«) . _ = /(<o + »*( «) 

a function which is by hypothesis finite. To this function there- 
fore the demonstration given in Art. 101 applies ; hence 

SQ + M -/'( *) + «*'(«) _ _ , ft*). 



therefore 



as before. 

Again, if the value ofv^ is infinite, that of -£-{ is zero, 

<t>(a) f(a) 

and, by the last result, 

f(a) f{aY 
hence, in this case, likewise 

t(a) #(a) ' 



Derivatives of Functions which assume an Infinite Value. 

104. When f(x) becomes infinite, for a finite value a of the in- 
dependent variable, f'(a) is likewise infinite. For, let b denote a 



Il6 EVALUATION OF INDETERMINATE FORMS. [Art. IO4. 

value of x so taken that /"(■*") shall be finite for x — b and for all 
values of x between b and a : then, as x varies from b to a, the 
rate of f(x) must assume an infinite value, otherwise/^) would 
remain finite. The value of x for which the rate is infinite must 
be a or some value of x between b and a ; that is, some value 
of x nearer to a than b is. Now, since b may be taken as. near 
as we please to #, the value of x for which the rate is infinite 

cannot differ from a. The expression for this rate is/'(;r) — , in 

dx 
which — may be assumed finite, therefore /"'(V) must be infinite 

when x = a ; in other words, f (a) is infinite when f(a) is infinite. 

105. It follows from the theorem proved in the preceding 
article that when a is finite the function obtained by the appli- 
cation of formula (3), Art. 101, takes the same form, — , as that 

00 

assumed by the original function. Hence, except when the 
given value of x is infinite, the application of some other process, 
either to the original function or to one of the substituted func- 
tions, is always requisite. Thus in the example, 



log (sin 2x)' 
log sin x 

by using the above formula we obtain 



log sin 2x 
log sin x _ 



2 cot 2X 
cot x . 



which takes the form — ; but the last expression is equivalent 



00 
sin x cos 2x~ 



to 2 

sin 2x cos x_ 

value unity. 



, and is therefore easily shown to have the 



§ XIV.] ALGEBRAIC FRACTIONS, X BEING INFINITE. WJ 

A I gcb j'ci ic Fractions, x being Infinite. 

106. A fraction of the following general form ; 

ax* + brf_ 4- c X* + • • - 
tf > a ' + //,r^' 4- £>*' -f- .... ' 

in which <r and ar' denote the highest exponents in the numera- 
tor and in the denominator respectively, takes the illusory form 

00 

- when x becomes infinite, provided a and a are positive. To 
determine the value of this fraction, we put it in the form 

x a {a + bx&- a + cx**~ a + • • •) 
x a '(a + b'xP'- a ' + <:>>'- a ' + • • y 

in which the exponents fi — a, y — a, fi' — a', etc., are all nega- 
tive. If now we make x infinite, all the terms within the marks 
of parenthesis, except the first, will vanish ; accordingly, the 
value of the fraction for this value of x is identical with that of 

a 

-x a ~ a . 
a 

When a > a' this value is infinite, when a < a it is zero, 

and when a — a' it reduces to the finite quantity -,. In either 

case, the value equals the ratio of the term of the highest degree in 
the numerator to that in the denominator. 
For example, 



4x h 4- 5>y 4 + 3-r 4- io" 
8^ 5 - 32 . 



The Form o . co . 

107. A function which takes this form may, by introducing 
the reciprocal of one of the factors, be so transformed as to take 



Il8 EVALUATION OF INDETERMINATE FORMS. [Art. I07. 



either of the forms - or , as may be found most convenient. 

o 00 
For example, let us take the function 

which assumes the above form when x — 00, n being positive. 

00 
In this case it is necessary to reduce to the form — . Thus — 



x~ n e x = 



nx h 



£ x 



n{n — i)x n 



, etc. 



By continuing this process, we finally obtain a fraction whose 
denominator is finite while its numerator is still infinite. Hence 
we have, for all finite values of n, 

X~ n £*] = 00. 
j 00 



The Form oo — oo. 

108. A function which assumes this form may be so trans- 
formed as to take the form - . Let the given function be 



I log (1 + x) 



x(\ + x) 



X' 



l 



which takes the form 00—00, since the second term is easily 
shown to be infinite. But 



r 1 _ iog(i + *) ~i _ x - (1 + x) log (1 + 

Lr(i 4- x) x 1 Jo x%i + x) 

_x — (1 + x) log (1 + x) 



^1 

Jo 



X' 



_ I - log (1 -f X) - I" 
2X 



§ X I V.] EXAMPLES. 1 1 9 

Examples XIV. 

Evaluate the following functions : 



i. when ^ = ^7r. 

sec $x 



2. ; —7 ', JC = CO 

cosec (w« ) 



log X , , 

3. -^- , (n > o), * = oo. 



tan x t 

4- i — 7 r~\ » ^ ~ 2 7r - 



log (* - ^tt) ' 
sec (§7Tx) 

Tog(i-x)' 



6. 


log COS (17TX) 


log (I -x) ' 


7- 


tan jc 
tan $x ' 


8. 


log ( i + x) 



JC = I. 



X = I. I. 



.# = J7T. 



3C = 00. 



9. I #* — 1 );;, ^=00. logtf. 

.x'' 2 — <r 7tx 4 

10. =—- tan — , x = a. . 

a 2a 7t 

11. x m (\ogx) n , (m and xl being positive), x = o. o. 

12. f-^sin — , ^= 00. 00. 



20 EVALUATION OF INDETERMINATE FORMS. [Ex. XIV. 



13. a * (1 — log*), 



nx . 1 
14. sec log — , 



J 5 



log tan nx 
I02: tan x f 



x = o. 



* = 1. 



* = o. 



o. 



I. 



log cot 



16. 



cot.x + log.*' 

17. sec x (x sin* — h n \ 

_ ■ / *\ 7T.r 

18. log 2 ) tan — , 

\ 0/ 2tf 

19. (1 — *) tan (Jtt*), 

20. log (* — a) tan (* — a), 



X = o. 



* = J7T. 



* = a. 



x = 1. 



* = #. 



* = «. 



o. 



tan 9 



Denoting the arc by 0, and multiplying by —7— {whose value, when 



x = a, is unity) we obtain — (a + x) 



:&-*) 



4a 

71 



2 2. Xt> 

sec m * 



23- 



24. * — jr log ( 1 4- 



%■ 



X = o. 



* = J7T. 



* = 00. 



00. 



o. 



Put X 



§ XIV.] EXAMPLES. 121 

X(f?* + «") _ 2f* I 



26. 



.V 



29. 


X I 


JC — 1 log X ' 


30. 


•* 


. g COt Xy 



3 1 - — » - -rr— 3- , - = o. 



I 




7T 




2JC' 2 


2x tan 


7tx f 


7t 




7T 




4x 


2.X 


(a™ 


+ 1)' 


7tX — 


1 

1 




7t 



X = o. 



2 
27. COt £#, JC = O. 



28. .# tan .* — \n sec x, # = \n. 



x = o. 



.* .#" sin jc 6 



o. 



7 
sin 3 x 6' 



**•• 



32. j r , x = o. \n 



33 - -ZT + ^eZTTy * = «• ** 

34. Prove that, when/(«) == 1 and $(#) = 1, 

iog/w = y>) 

log <j>(a) <f>'(a) ' 

35. Prove that, when/(«) and <j>(a) = o, 

\og/U) = 
log 0(a) 

provided that ,,, > is neither infinite nor zero. 



122 EVALUATION OF INDETERMINATE FORMS. [Art. IO9. 

XV. 

Functions whose Logarithms take the Form oo-o. 

109. In the case of a function of the form w, we have 

log u v —v log u. 

The expression v log u takes the illusory form o • 00 in two 
cases : first, when v — o and log u — 00 ; and secondly, when 
v = 00 and log u — o. 

Log u is infinite when u = o, and also when u = 00 ; there- 
fore the first case will arise when the original function takes 
one of the forms oo° or o°. 

Log u = when u = 1, therefore the second case will arise 
when the original function takes the form 1 °°. 

Hence functions which take either of the three illusory forms. 

oo°, o°, or I 00 , 

may be evaluated by first evaluating their logarithms, which 
take the form o • 00. 

It is to be noticed however that o 00 and 00 °° are not illu- 
sory forms, since their logarithms take the form 00 (=j= 00). 

The Form i c °. 

110. As an illustration of this form, we take the function 

H — . which assumes the form 1 °° when x — 00. Denot- 

xJ 

ing this function by u, we have 

a 



log u = *log( 1 ■+--) = 



XJ I 



the last expression assuming the form — when x — 00. 



I XV.] T//£ FORMS »°, o°, AND i". 1 23 

In eval this logarithm, it is convenient to substitute 

z for — ; then 



los" u — 



log* 1- 



■, 



since, when x = 00, ar = a Taking derivatives, we have 

. logi I - OS 1 

o ~ » 



Hence # _ = 1 = £• 

III. If a — I, we have 



\ * J DC 



that is. as ,r increases indefinitely, the limiting value of the func- 
tion r is c. The Napierian base is often defined as the 

limiting value of this function, or, what is the same thing, by 
formula 

«r=(l -xu] o . 

The Form o°. 

112. The function -.r" r ]_, by the aid of which many functions 
of similar form may be evaluated, will serve as an illustration 
of the form o : . 

Let u = x x \ 

then log u = x logx, 



124 EVALUATION OF INDETERMINATE FORMS. [Art. 112. 



and log; u 



t> 



_ log x 



o; 



therefore x ' v \— £p = *• 

The value of a function which takes the form 0° is usually- 
found, as in the above example, to be unity. This is not, how- 
ever, universally true, as the function 

a + x 

x \o%x 

(one of those earliest adduced for this purpose*) will show. 

This function takes the form o°, when x — o; but since its 
logarithm reduces to a + x, its value when x = o is e a . 



Functions having two limiting vahies for a single 
value of x. 

113. An exponential function may have two limiting values 
corresponding to that value of x for which the exponent be- 
comes infinite ; when this is the case the function is said to be 
discontinuous. 

Thus the function e x increases without limit when x is posi- 
tive and approaches zero ; but when x is negative, denoting its 
numerical value by x ' , we have 

a* = a ■*', 

which becomes zero when x vanishes. These results may be 
expressed thus — 

£*] =oo, but £~-'] o =o, 

it being understood that x in the first expression is positive 

* See Crelles Journal, vol. xii, p. 293. 



§ XV.] DISCONTIGUOUS FUNCTIONS. 1 25 

until it reaches the value zero, and that in the second expres- 
sion x is positive until it reaches zero. The curve y — e x is 
traced in Art. 247. 

114. The same peculiarity belongs also to the function 

1 
u -- x" e*, 

n having any finite value. For, supposing x to be positive, 
and denoting by n the value of the function when x vanishes, 
u is evidently infinite when n is negative, but takes the form 
o • 00 when n is positive. To evaluate u in this case we put 

1 



; whence 

Uo = x n €^] Q = 



This function has been shown, in Art. 107, to have an infinite 
value ; therefore, we have for all values of n (x being positive) 

U Q = CO. 

Now, when x is negative, let x — — x', then 
u = (— x) n e"< 

Denoting the value which this function assumes when x is 
positive and vanishes by u' Q , we have 



u = 



{-x 



n ex ' 



The denominator of this fraction is infinite for all values of n 
by the demonstration given above ; hence we have, when x is 
negative, for all values of n 



0. 



126 EVALUATION OF INDETERMINATE FORMS. [Ex. XV. 



Examples XV. 



i. (cos*) 001 '-*, 

I 

/tan x\~2 



when x = o. 



3. (cos^) cosec2 ^, 



.* = o. 



X = o. 



JC = o. 



V*. 



?/3- 



5. (tan*) tan8 *, 

6- (-=) («>o), 



i*. 



7. (1 —x)', 

8. (sin*) 5602 -^ 

9. (cotxY in % 
Solution: (cot#) sin! 

10. (sin^) tana; , 

11. (sin^) tanx , 



x = o. 



(cos*) 8l "'] c 

(sin ^) biu,r ]c 



a- = o. 
1. (fe^r/. 112.) 



jc = o. 



1. 



12 ylogsin_ir 



13. (sinjc) lo s tan ^, 

14. x xa (a > o), 

(* + #)* 



n 
x = — . 

2 


1. 


# = 0. 


£ rt . 


^ = 0. 


£" 2 . 


# = 0. 


I. 



I?. (# 2 ) log (-* + log cos -*■) 



JC = o. 



§ XV.] 



EXAMPLES. 



127 



n — ix 



16. x 

17. .v € 

18. (cos w*)^ 8 , 

I 

20. (1 ± x)- v , 

21. A7 m (sin^) tan ' 

\2 Sin 2X, 

(uf — 1) (a sin x — sin ^^) rt 

2 2. — ; ' — - — 

x n sin x (cos x — cos ax) n ' 
1 

( 1 — .*) x r , . . . . 
23 " *»(*- i) L^ ~ ^ g ^ ~~ ^ + * 

(1 4- *) ; — f 

2 5- Z 



when x = 1. 



1. 



^ = 0. f -*»/«» 



.%• = 00. 



7T 



J log;;/. 



#) + x 



X — I. 



J? = o. 



I 

2£ 



— I. 



Solution . — 

Denoting (1 + -*) 7 by y } and observing that, by Art. in, when 

x — °> y — £ 5 ^ is evident that the function takes the form - ; 

o 



hence, on applying the usual process, its value is 
ing, we have 



dx 



Differentiate 



dx 



r 1 iog(i +x) ~\ _ 

'[_x{i +x) x 2 J **> 



28 EVALUATION OF INDETERMINATE FORMS. [Ex. XV. 



in which z denotes the function evaluated in Art. 108, hence z = — -§-. 
Therefore, 



26. 



(1 + x)» — s~ 



x(i + x) x — esmx) 



dx 



when x = o. 



Solution : — 

Using the notation and results of the preceding example, we have 

-1 y +x~ fcos^n 

xy — £ sin x _ dx 

X 2 — lo 2JC 



_ jv — £ cos X 



2X 



" dx 



dx 



£ sin x' 



£ _ £ 

4~ 2 



27. 2 being denned as in Example 25, prove that 



dx 



28. 



* £# 

(1 + x) x — e + — 

V ' 2' 



when x = o. 



£/><? //£<? notation and results of Examples 25 #/^ 27. 

19. — (1 + #)* log (1 + #) L when # = o. 

jy — £ x~ x log (1 + #) 



Il£ 
24 



H 



/>#/ /« the form 



{See Example 25.) 



§XV.] 



EXAMPLES. 



129 



30. If y = p , give the value of y, and also that of ~- , when 



I + £■ 



dx 



x approaches zero from the positive and from the negative side. {See 
Art. 113.) 

Result, (positive side) : y z= o ; ~- = o ; 

ax 

(negative side) : y = 1 ; — = o . 



XVI. 



Indeterminate Forms of Functions of Two Variables. 

115. When a function of two independent variables takes an 
illusory form, its value is in general indeterminate. For ex- 
ample, the function 

x 2 — ?yxy + x 
f- — xy + 1 



takes the form -, when x — 2 andjy = 1. Employing the usual 
method, we obtain 



x* — ^xy + x~\ 
/ — xy + 



2x-3y + 1 -$x 



I J 2 ,x 



dy~ 

~dx 



f \ dy 



dx_ 



The value of this expression depends upon that of -5-, and hence 
is really indeterminate when x and y are independent ; but, when 
x and y are connected by an equation, -J- has a definite value 



130 EVALUATION OF INDETERMINATE FORMS. [Art. 1 1 5. 

for the given values of x andj, and consequently the given 
function has also a definite value. 

116. In certain cases, the value of a function of two variables 

d v 
which takes an illusory form is independent of that of -j- ; thus, 

&x 

\o<gx + log y 

x + y — 2 

takes an indeterminate form, when x — I and y— I. Evalu- 
ating this expression we obtain 



x y dx 
dx 



which is independent of -~- 



;],., 



dx.\ i, x 



7^£ Evaluation of Derivatives of Implicit Functions. 

117. When y is an implicit function of x, its derivative is an 
expression involving both x and y ; hence, if it takes an inde- 
terminate form for given values of x and y, the expression ob- 
tained by the usual process of evaluation will involve the re- 
quired derivative. Thus, given 



y 4- 3#y — 2axy — ax* = o, . . . . (i) 

dy 
ale 

values which satisfy equation (i). In this case 

dy 2ay 4- 2ax 



to find the value of -f-, when x — o and y = o, these being 



dx $y* -f 6ay — 2ax 



§ XVI.] DERIVATIVES OF IMPLICIT FUNCTIONS. 



131 



which takes the form - for x — o and y = o. Evaluating, we 
o 

obtain 



dy 
dx 



'_Jo, o 



2d 



dy 
~dxA 



+ 2a 



dy 

dxAo 



+ 1 



6a d -l 
ax. 



dy 

~ 2a 3 si 



an equation involving the required derivative in both members. 
Clearing this equation of fractions, we obtain the quadratic 



dy 

3 dxJ 



dy 



o, o ax 



= I, 



whence we have 



dy 
dx_ 



= 1, and -f 
dxj< 



This result shows that, regarding (1) as the equation of a 

d y 
curve, there are two values of —-, or tan ^, at the origin, indi- 
cating that the curve passes twice through the origin. 

(18. When x — o and y — o are simultaneous values of x and 

y O 

y. the ratio - takes the form - , and by the formula for evalua- 
-" x o J 

tion we have 



dy' 
dx. 



When the value of 



dy 

dxJc, 



is required, we may therefore employ 



the expression 



y 



which often admits of evaluation by an 



132 EVALUATION OF INDETERMINATE FORMS. [Art. 1 1 8. 

algebraic process. Thus dividing equation (1) of the preceding 
article by x 2, we have 

y 2 y 2 y 

y J — + $a •<- - — 2a <- — a — o, 
x 1 x* x 



in which if we put y = o, and assume the value of — 
finite, we obtain 



to be 



y 



y 

2 - 



I = o. 



This quadratic is the same as the one employed in the preced- 
ing; article to determine -^- . 

It is obvious that in this process all the terms in the given 
equation, except those lowest in degree, disappear when we put 
x = o and y = o ; hence the same result may be obtained by 



y 



putting the terms of lowest degree equal to zero. When 



is 

infinite the same reasoning is applicable, since we may in this 
case emi 



iploy the reciprocal — . 



v 

In the given example we find two values of — 

x. 



y- 

because the 



terms employed are of the second degree. 

Examples XVI. 

1. The variables x and y being connected by the equation 

2(1 — x + y) — 4/ + 3* 4 = o, 

show that x — o and y — 1 are simultaneous values, and find the cor- 
responding value of 

f + x % - Sx 2 + x - 1 _ _ 

o . Result o. 

xy — 4X' 



§ XVI.] EXAMPLES. 133 

2. The relation between x sandy being expressed by the equation 

3* + 4/ - 24 (x + y) + 37 = o, 

show that x — 1 and y = 2 are simultaneous values, and find the cor- 
responding value of 



y- x-- 3 

/// this example, on substituting the numerical value of -j- , the function 
again takes the indeterminate form ; it is therefore necessary to substitute 
the value of — in terms of x and y, and to repeat the process. 

3. The relation between x and j* being expressed by the equation 

y- + x~ — 2£x — t~ = o, 

show that x = o and y = f are simultaneous values, and find the cor- 
responding value of 

e x — logy e — 1 

/ — log (e — x) ' £ + 1 * 

n ft {fy 

4. Given y x = x y ; find the values of -f- when x = e and 

y dx 

y—e. ± 1. 

5. Given x* — 3a xy + y 3 + a? = o ; 

show that jc = # and/ = tf are simultaneous values, and find the cor- 
responding values of -f- . J[i + /(— 3)]. 

6. Given a: 3 — $xy + y 3 = o, 

find the values of h— when x = o and j — o. o and 00. 



134 EVALUATION OF INDETERMINATE FORMS. [Ex. XVI. 

7. Given y* + -^a'y' — j\a'xy — cfx 1 = o ; 
find the values of -f- when x = o and 7 = 0. 

M« ± V7)- 

8. Given x 3 — axy + 2 .# — ay 1 + 20 s / — # 3 = o ; 

show that x = o and y = a are simultaneous values, and find the cor- 
responding values of -3— . 

or o, and — 1. 

9. Given y* — 960^* + 1000 s * 8 — x* = o ; 

dy 
find the values of — corresponding to x = o and 7 = o. 



10. Given y* — x* — ^ay 3 + 5a V + 20 s * 8 — 20 3 j/ = o ; 

show that x = o and 7 = are simultaneous values, and find the cor- 
responding values of -f- . 

dx ± Y2. 

11. Given x* — 2a y 3 — 30 s j/ 2 — 20V + a* = o ; 
find the values of -j- when jy = o, also when x = o. 



0j~| , 2 ^ 



-] = ± f 

J±«,o 



0* +«.o 0* 



4/3 



±J|/6, and -^ 



= o. 



12. Given y 3 — a(x + 0) (x + y) = o ; 

find, by the method of Art. 118, the value of -j- \ . 

dX Jo.o 



dxj , 



13. Given a: 4 + ax*y — ay 3 = o ; 

dy~\ 

_lo, o 



find the values of -~ when * = o 
dx 



o or ± 1. 
dx 



14. If _v =x (x — 1) log (# ± Vjp), 



§ XVI.] 



EXAMPLES. 



35 



dy 
find, by the method of Art. 118, the value of ~f- when x = o ; also 

ax 

its values when x = i, by substituting x for x — i. 






A' 



oo and log 2. 



15. Prove by putting^ = x/(x), and comparing the values of 



dy_ 
dx 



as found by differentiation and by the method of Art. 118, that if 
/(o) is finite x/(x) = o, when x = o, even though /'(o) is infinite. 



CHAPTER VI. 

Maxima and Minima of Functions of a -Single 
Variable. 



XVII. 

Conditions Indicating the Existence of Maxima 
and Minima. 

119. If, while the independent variable increases continu- 
ously, a function dependent on it increases up to a certain 
value, and then decreases, this value of the function is said to 
be a maximum value. In other words, a function f(x) has a 
maximum value corresponding to x = a, if, when x increases 
through the value a, the function changes from an increasing 
to a decreasing function. 

Since f\x) is positive, when f{x) is an increasing function, 
and negative when it is a decreasing function ; it is obvious 
that if f(a) is a maximum value of f{x),f'(x) must change sign, 
from + to — , as x increases through the value a. 

On the other hand, a function is said to have a minimum 
value for x — a, if it is a decreasing function before x reaches 
this value and an increasing one afterward. In this case, f'(x) 
changes sign from — to +. 

120. The derivative f\x) can only change sign on passing 
through zero or infinity. Hence a value of x, for which f(x) 
is a maximum or a minimum, must satisfy one of the two follow- 
ing equations : • 

f\x) = o and f'( x ) — °°* 



>? XVI I.J MAXIMA AND MIX I MA. 1 37 



The required values of x will therefore be found among the 
roots of these equations. 

The case which usually presents itself, and which will there- 
fore be considered first, is that in which the required value of 
x is a root of the equation fix) = o. 

121. As an illustration, let it be required to divide a number 
vito two sucJi parts that tJie square of one part multiplied by the 
cube of the other shall give the greatest possible product. 

Denote the given number by a, and the part to be squared 
by x ; then we have 

f{x) = x- {a - x)\ 

It is evident that a maximum value of this function exists ; 
for when x = o its value is zero, and when x = a its value is 
again zero, while for intermediate values of x it is positive ; 
hence the function must change from an increasing to a decreas- 
ing function at least once, while x passes from the value zero to 
the value a. 

Taking the derivative of this function, the equation 

/'(*) ='o 

is in this case 2x(a — xf — 3.1' 2 (a — xf — o, 

or x(a — x)~ (2a — 5,1-) = o. 

O and a are roots of this equation ; but, as we are in search of 
a value of the function corresponding to an intermediate value 
of x, we put 

2a — $x = o, 

and obtain x = \a. 

The corresponding value of the function is -^j%\d% the maxi- 
mum value sought. 



MAXIMA AND MINIMA. 



[Art. 122. 



Maxima and Minima of Geometrical Magnitudes, 

122. When the maximum or minimum value of a geometri- 
cal magnitude limited by certain conditions is required, it is 
necessary to obtain an expression for the magnitude in terms of 
a single unknown quantity, such that the determination of the 
value of this quantity will constitute the solution of the prob- 
lem. For example : let it be required to determine the cone of 
greatest convex surface among those which can be inscribed in a 
sphere whose radius is a. 

Any point A of the surface of the 
sphere being taken as the apex of 
the cone, let the diagram represent 
a great circle of the sphere passing 
through the fixed point A. 

If we refer the position of the 
point P to rectangular coordinates, 
and take C as the origin, the required 
cone will evidently be determined 
when x is determined. We have 
now to express the convex surface 
5 in terms of x. 
The expression for the convex surface of a cone gives 

S=7TyV[/ + (a+xyi (i) 

in which the unknown quantities x and y are connected by the 
equation of the circle 

**+y = d (2) 

Substituting the value of y, we have 




Fig. 



reducing, 



5 = n V(a 2 - x*) Y{2a" + 2ax), 
S = 7i V(2a) {a + x) V(a — x). 



(3) 



§ XVII. 



CEO ME TRICAL MA GNITUDES. 



139 



Since the factor n 1/(20) is constant, we are evidently re- 
quired to find the value of x for which the function 

fix) = (a + x) V{a - x) 

is a maximum. The equation f\x) = o is, in this case, 

V{a - x) — = o ; 

v } 2 Via — x) 



whence 



ha. 



The altitude of the required cone is therefore f#. Substi- 
tuting this value of x in equation (3), we have 

S=%Vyna\ 

the maximum value required. 

I23. As a further illustration, let it be required to determine 
the greatest cylinder that can be in- 
scribed in a given segment of a pa- 
raboloid of revolution. 

Let a denote the altitude, and b 
the radius of the base of the seg- 
ment. The equation of the gener- 
ating parabola is of the form 



y 



4cx. 



Since (a, b) is a point of the curve, 
we have the condition 

b\ = 40a ; 
eliminating 4c, the equation of the curve is 

. b* 




Fig 11. 



y = — x. 
a 



(1) 



I40 MAXIMA AND MINIMA. [Art. 1 23. 

The volume V of the cylinder of which the maximum is re- 
quired is expressed by 

V — 7ty\a — x\ 

fr 2 
or, by equation (1), V — n — x{a — x). 

Hence we put fix) = ax — x*, 

and the condition f\x) — gives 

x — \a. 

Consequently a — x, the altitude of the cylinder, is one half the 
altitude of the segment. 

Examples XVII. 

1. Find the sides of the largest rectangle that can be inscribed in 
a semicircle of radius a. The sides are a 4/2 and \a 4/2. 

2. Determine the maximum right cone inscribed in a given sphere. 

The altitude is four thirds the radius of the sphere. 

3. Determine the maximum rectangle inscribed in a given segment 
of a parabola. 

The altitude of the rectangle is two thirds that of the segment. 

4. Find the maximum cone of given slant height a. 

The radius of the base is \a \/6. 

5. A boatman 3 miles out at sea wishes to reach in the shortest 
time possible a point on the beach 5 miles from the nearest point of 
the shore ; he can pull at the rate of 4 miles an hour, but can walk at 
the rate of 5 miles an hour ; find the point at which he must land. 

Express the whole time in terms of the distance of the required point 
from the nearest point of the shore. 

He must land one mile from the point to be reached. 



§ XVII.] EXAMPLES. HI 

6. If a square piece of sheet-lead whose side is a have a square cut 
out at each corner, find the side of the latter square in order that the 
remainder may form a vessel of maximum capacity. 

The side of the square is \a. 

7. A given weight is to be raised by means of a lever weighing n 
pounds per linear inch, which has its fulcrum at one end, and at a 
fixed distance a from the point of suspension of the weight w ; find the 
length of the lever in order that the power required to raise the weight 
may be a minimum. /2aw 

v—- 

8. A rectangular court is to be built so as to contain a given area 
e*, and a wall already constructed is available for one of the sides ; 
find its dimensions so that the least expense may be incurred. 

The side parallel to the wall is double each of the others. 

9. Determine the maximum cylinder inscribed in a given cone. 

The altitude of the cylinder is one third that of the cone. 

10. Prove that the rectangle with given perimeter and maximum 
area is a square ; also that the rectangle with given area and minimum 
perimeter is a square. 

n. Find the side of the smallest square that can be inscribed in a 
square whose side is a. 

Take as the independent variable the distance between the angles of the 
two squares. £a 4/2. 

12. Inscribe the maximum cone in a given paraboloid, the apex of 
the cone being at the middle point of the base of the paraboloid. 

The altitude of the cone is half that of the paraboloid. 

13. Find the maximum cylinder that can be inscribed in a sphere 
whose radius is a. The altitude is %a ^3. 

14. Through a point whose rectangular coordinates are a and b draw 
a line such that the triangle formed by this line, and the coordinate 
axes shall be a minimum. 

The intercepts on the axes are 2a and ib. 



142 MAXIMA AND MINIMA. [Ex. XVII. 

15. A high vertical wall is to be braced by a beam which must pass 
over a parallel wall a feet high and b feet distant from the other ; 
find the length of the shortest beam that can be used for this purpose. 

Take as the independent variable the inclination of the beam to the 
horizon. 3. 

U 3 + b A ) . 

16. From a point whose abscissa is c, on the axis of the parabola 
y 1 = ^ax, determine the shortest line to the curve. 

The abscissa of the required point on the curve is c — 2a. 

17. Determine the greatest rectangle that can be inscribed in the 
ellipse 

x 1 y~ 

a 

The sides are a\/2 and £4/2. 

18. A cylinder is inscribed in a cone whose altitude is a, and the 

radius of whose base is b ; determine the cylinder so that its total surface 

shall be a maximum, and thence show that there will be no maximum 

when a< 2b. _. . . . . a?— 2ab 

1 he altitude is —, rr . 

2{a— b) 

19. Determine the cone of minimum volume described about a given 
sphere. The height is twice the diameter of the sphere. 

20. A sphere has its centre in the surface of a given sphere whose 
radius is a ; determine its radius in order that the area of the surface 
intercepted by the given sphere may be a maximum. 

The area of a zone is measured by the product of the circumference of 

a great circle into the altitude of the zone. 4 

■%a. 

21. Find the point, on the line joining the centres of two spheres 
whose radii are a and b, from which the greatest amount of spherical 
surface is visible. 

The distance between the centres is divided in the ratio a* : b 2 . 

22. In a given sphere, determine the inscribed cylinder whose en- 
tire surface is a maximum. 



§ XVII.] EXAMPLES. 143 

Solution : — 
Using the notation of Art. 122, we find 

f(x) = a~ — x' + 2x V(<i~ — x') ; 

2 

whence f\x) = — 2x + 2 V(a' 2 — x~) 



vw-xy 

and /'(*) = o gives 

x V(a* - x") = a' - 2X* (1) 

Squaring, we have 

$x — 5« x 4-^=0, 
the roots of which are 

a* = a\\ ± J Vi) : 

but since the radical in equation (1) must be positive, we must have 
x 2 < Jtf 2 ; hence the altitude, 2x, of the cylinder is 

23. In a given sphere determine the inscribed cone whose entire sur- 
face is a maximum. 

The altitude of the cone is — (23 — 4/17). 

16 

24. A cylindrical trough is constructed by bending a given sheet of 
tin ; its breadth being denoted by 20, find the radius of the cylinder 
when the capacity of the trough is a maximum. 

Solution : — 

Let x denote the radius of the cylinder ; then - will be the measure 

x 

of the half-angle of the circular segment which constitutes a section of 

the trough. The area of the section will be expressed by 



Hence f\x) = o gives 



a . a 
ax — x cos - . x sin - . 

x x 



a a .a 

cos - a cos x sin - I =0, 

X\ X X 



a 




a a 


cos -- — o, 


or 


tan - = - 


X 




X X 



144 MAXIMA AND MINIMA. [Ex. XVII. 



therefore 

x 

There is evidently a maximum between - = o and - =7t. - = — is a 

root of the first of the above equations, and since it is the only root of 
either equation between these limits, it must correspond to the maxi- 
mum sought. Hence the section is a semicircle. 

25. The illumination of a plane surface by a luminous point being 
directly as the cosine of the angle of incidence of the rays, and in- 
versely as the square of its distance from the point ; find the height 
at which a bracket-burner must be placed, in order that a point on 
the floor of a room at the horizontal distance a from the burner may 
receive the greatest possible amount of illumination. 

The height is —7-. 



XVIII. 

Methods of Discriminating between Maxima and 

Minima. 

124. When the existence of a maximum or a minimum cor- 
responding to a particular root a of the equation fix) = o is 
not obvious from the nature of the problem, it is necessary to 
determine whether fix) changes sign as x passes through the 
value a. 

If a change of sign does take place we have, in accordance 
with Art. 1 19, a maximum if, when x passes through the value 
a, the change of sign is from + to — ; that is, if fix) is a de- 
creasing function, and a minimum if the change of sign is from 
— to +, in which case f(x) is an increasing function. 

125. In many cases we are able to distinguish maxima from 
minima by examining the expression for f'(x), as in the fol- 
lowing examples. 



§ XVIII.] METHODS OF DISCRIMINATING. 



145 



Given 



whence 



/(*) = 



log* 



-., s \o$fX — I 



(log*) 1 ' 

f'(x) = gives log;r=i, or ;r = f. 

Since log* is an increasing function, it is obvious that, as x in- 
creases through the value £,f'(x) increases ; it therefore changes 
sign from — to + , and consequently /(e) is a minimum value 

of /(*). 

126. If f'(x) does not change sign we have neither a maxi- 
mum nor a minimum ; thus, let 

f(x) = x — sin*, 

whence f\x) = 1 — cos*. 

In this case f'(x) becomes zero when x — 2ii7t, n being zero 
or any integer, but does not change sign, since 1 — cos* can 
never be negative ; consequently /(*) has neither maxima 
nor minima values, but is an increasing function for all values 
of x. 



Alternate Maxima and Minima, 

127. Let the curve 

y=f(x) 

be constructed, and suppose it to take the form represented in 
Fig. 12. There is a maximum value of 
fix) at B, another at D, and minima 
values occur at A, at C, and at E. 

It is obvious that in a continuous por- 
tion of the curve maxima and minima 
ordinates must occur alternately, and 
must separate the curve into segments 
in which the ordinate is alternately an 
increasing and a decreasing function ; hence, if f(x) has maxi- 



I46 MAXIMA AND MINIMA. [Art. 1 27, 

ma and minima values, they must occur alternately unless infi- 
nite values of the function intervene. It is also evident, with 
the same restriction, that a maximum is greater in value than 
either of the adjacent minima, but not necessarily greater than 
any other minimum ; thus, in Fig. 12, the maximum at B is 
greater than the minima at A and C, but not greater than 
that at E. 

128. As an illustration let us take the following function in 
which it is easy to discriminate between the maxima and min- 
ima values. 

f(x) = x{x + aj (x — a)\ 
Whence, 

f'(x) = (x + af (x — a) 5 + 2x(x + a) (x — a) 3 + $x(x + a) 2 (x — a)\ 

— (x + a) {x — a) 2 (6^ 2 + ax — a'). 

a and — a are evidently roots of f\x) = o ; the roots derived 
by putting the last factor equal to zero and solving are — \a 
and \a. Hence f\x) can be written in the form 

f\x) = 6(x + a) (x + la) (x — ia) (x — a)\ 

in which the factors are so arranged that the corresponding 
roots are in order of magnitude. 

When x < — a, f\x) is negative, and, if we regard x as in- 
creasing continuously, f'(x) changes sign when x — — a, when 
x = — %a, and again when x = ^a, but not when x = a. 

Since fix) is at first negative it changes sign from — to + 
when it first passes through zero, that is when x — —a; the 
corresponding value of f{x) is therefore a minimum. Accord- 
ingly the value of f(x) corresponding to the next root x — — \a 
is a maximum, and that corresponding to x = \a is another 
minimum ; but there is neither a maximum nor a minimum 
corresponding to x = a. 



§ XVIII.] ALTERNATE MAX/MA AXD MINIMA. I47 

129. When the function is continuous as in the above ex- 
ample, that is, does not become infinite for any finite value of 
.r, it is always easy to determine by examining the function 
itself whether the last, or greatest value of x in question, gives 
a maximum or a minimum. Thus, in the above example, f{x) 
evidently increases without limit as x increases without limit ; 
therefore, the last value must be a minimum. 



The Employment of a Substituted Function. 

130. Since an increasing function of a variable increases and 
decreases with the variable, such a function will pass from a 
state of increase to a state of decrease, or the reverse, simulta- 
neously with the variable ; that is, it will reach a maximum or 
a minimum value at the same time with the variable. 

This fact often enables us to simplify the determination of 
maxima and minima by substituting an increasing function of 
the given function for the given function itself. For example, 
if we have 

f{x) = V(b°- + ax) + V(b* - ax), 

we may with advantage employ the square of the given func- 
tion. The square is 

2b 2 -j- 2 V(b A - a~x% 

which is obviously a maximum when x = o, and, since the square 
of a positive quantity is an increasing function, we infer that 
fix) is likewise a maximum for the same value of x. 

131. It must however be observed that, if the original func- 
tion assumes the value zero, the square will likewise become 
zero, and, since a square cannot become negative, this value 
(zero) will be a minimum value of the square even when the 



H 8 MAXIMA AND MINIMA. [Art. 1 3 1. 

original function has neither a maximum nor a minimum value. 
Thus let the given function be 

x — 2a 
Employing the square, we put 

(X — 2af 



whence 

ft \v_ 2 (^ 2 ~~ ^ 2 ) ( x ~~ 2a ) ~ 2x ( x ~~ 2a Y _ 2 ( x — 2 ^) ( 2ax ~ a *) 
J W ~~ [x 2 - a 2 ) 2 ~~ ~~ (x 2 ~aj " 

fix) changes sign when x = 2a and when x = \a. 

The former value of x gives f(x) — o which is a minimum of 
this function ; but, when x passes through the value 2a, the 
original function evidently changes sign, and is therefore neither 
a maximum nor a minimum. The other value of x, %a, gives 
an imaginary value to the original function ; we therefore con- 
clude that the given function has neither maxima nor minima 
values. 

132. A decreasing function of the given function may also 
be employed ; but, in this case, since the substituted function 
decreases with the increase of the given function and increases 
with its decrease, a maximum of the substituted function indi- 
cates a minimum, and a minimum indicates a maximum of the 
given function. 

Thus, if we have 

/<*) = ^-^+1 - 

the reciprocal may be employed. The reciprocal of this func- 
tion is 

X* — XX + I ,1 

3 = x - 3 + - ; 

X X 



§ XVI 1 1, j SUBSTITUTED FUNCTIONS. H9 



whence, taking the derivative, we obtain 

i x 2 — i 

1 i = 2 — > 

x~ x~ 

which vanishes when x = ± I. 

Since x* is an increasing function when x is positive, this deriv- 
ative is evidently an increasing function when x =* i. The re- 
ciprocal is therefore a minimum for this value of x, and conse- 
quently /(i) is a maximum value of /"(■*)• In a similar manner 
it may be shown that f{— i) is a minimum. 

Examples XVIII. 

Determine the maxima and minima of the following functions : 

i. f (x) = x x . A min. for x 



i 



logJ£ 



2. fix) = . A max. for x — £». 

., . (a — xY . . 

7. / (.*) = — . A mm. for x = ±a. 

a — 2x 

4- fW = —77 — r • A max. for x = — i T V 

* y w 1/(4 + 5*) lT 

5- /(•*) — sin 2x -- j:. A max. for x ~ nn + \n \ 

a min. for x — ^ n — i7T. 

6. /(#) = 2# 3 + 3^:" 2 — $6x +12. A max. for x — — 3 ; 

a min. for # = 2. 

7. /(#) = .# 3 — 3^ 2 —9^+5. A max. for x = — 1 ; 

a min. for 3: = 3. 

8. /(#) = 3-r 5 --i25# 3 + 2i6ox A max. for x~—^ and #=3 ; 

a min. for x=—$ and #=4. 



I$0 MAXIMA AND MINIMA. [Ex. XVIII. 

9< f(* x ) — ^ + c (# ~ #) • Neither a max. nor a min 

10. /(#) = (x — i) 4 (# + 2) 3 , A max. for # = — \ 

a min. for x = i 

ii. /"(■#) =■(#-* 9) 5 (# — 8) 4 . A max. for >r = 8 

a min. for ^ = 8f 

12, Find the maximum and minimum ordinates of the curve 

y = x" + 2# 3 — I2^C 2 — ^QX — 34. 

A min. for x = 2 J. 
,. v sin 2 ,* 

I^. /(jC) = —77 r. 

^(5-4 cos X) 
To discriminate between maxima and minima observe thatfio) = o 
and that f{x) cannot become negative. See also Art. 127. 

A min. for x = o ; 
a max. for x = cos -1 ^(5 — 4/13). 

14. f(x) == t — . A min. for x = £. 

„ . . Max. for x = 1 ; 

I 5- /(*) = — 1 — -7—; — • See Art. 132. . r 

*> y v «x 2 — bx + min. for x — — 1. 

16. /(#) = (a I_1 ) 2+i - 

Min. for x = — -£- (0 being positive). 

17' /(*) = (i 4-**)( 7 -*) 2 . 

*&?/z>£ by putting x = z*. For method of discriminating between max- 
ima and minima, see Art, 129. Min. for 3; = o ; and x = 7 ; 

max. for # = 1. 

r8. f(x) = 5X 6 + i2jc 5 — 15.* 4 — 40JC 3 + 153; 2 4- 60^ + 27. 

Min. for x = — 2. 

19. /(#) = x e — 6x i +' 4^ 3 + 9^ 2 — 12JC + 3. 

Min. for x = — 2, and x = 1 ; 
max. for x — 1. 



I 


— 


JV + 


* 2 


I 


+ 


x — 


* 2 - 



§ XVIII.] EXAMPLES. 151 

20. The top of a pedestal which sustains a statue a feet in height is 
b feet above the level of a man's eyes ; find his horizontal distance from 
the pedestal when the statue subtends the greatest angle. 

When the distance = V[b(a + b)~\. 

21. It is required to construct from two circular iron plates of radius 
a a buoy, composed of two equal cones having a common base, which 
shall have the greatest possible volume. 

The radius of the base = \a. 

22. The lower corner of a leaf of a book is folded over so as just to 
reach the inner edge of the page ; find when the crease thus formed is 
a minimum. 

Solutio?i . — 

Let j- denote the length of the crease, x the distance of the corner 
from the intersection of the crease with the lower edge, and a the 
width of the page. 

By means of the relations of similar right triangles, the following 
expression is deduced : 

x Vx 
} '~ V(x-ia)' 
Whence we obtain 

x-%a, 

which gives a minimum value of y. 

23. Find when the area of the part folded over is a minimum. 

When x = § a. 



XIX. 

The Employment of Derivatives Higher than the First. 

133. To ascertain whether f\x) is an increasing or a de- 
creasing function, (and thence whether f(x) is a minimum or a 
maximum), it is frequently necessary to find the expression for 
its derivative, fix). Now, \if"(d) is found to have a positive 
value, it follows that_/"(V) is an increasing function when x =. a, 



152 MAXIMA AND MINIMA. [Art. 1 3 3. 

and, as was shown in Art. 124, that/(V) is a minimum. On the 
other hand, if we find that f"(a) has a negative value, it follows 
that /'(■*) is a decreasing function, and that/"(tf) is a maximum. 
To illustrate, let 

f(x) — 3** — i6x 3 — 6x' 4- 12, 

then f\x) = I2x 3 — 48^ — \2x. 

The roots of f\x) = o are ^ = o, and x — 2 ± 4/5. 

In this case f"(x) = 36-r 2 — 96^ — 12, 

hence /"(°) = — .12; 

/"(V) is therefore a maximum when x = o. 

It is unnecessary to find the values oif"(x) for the other 
roots ; for, since the function does not admit of infinite values, 
the maxima and minima occur alternately. The root 2 — V 5 
being negative and 2 -j- V$ positive, the root zero is intermediate 
in value, and therefore both the remaining roots give minima. 

134. If fix) contains a positive factor which cannot change 
sign, this factor may be omitted ; since we can determine 
whether f\x) increases or decreases through zero by examin- 
ing the sign of the derivative of the remaining factor. Thus, if 



i + x" J w (i+xy 

Since 7 ^-r is always positive, we have only to determine 

(1 -f x 2 ) 2 J r J 

whether the factor I — x 2 changes sign. Denoting this factor 
by v, and putting v — o, we have 

x= ± 1. 

NOW -r- = — 2X 

ax 

which is negative for x—\ and positive for x— — 1. These 



§ XIX.] EMPLOYMENT OF SECOND DERIVATIVES. 153 



roots, therefore, give respectively a maximum and a minimum 
value oi f(x). 

135. There may be roots of the equation f'(x) — o which 
correspond to neither maxima nor minima, since it is a condi- 
tion essential to the existence of such values that fix) shall 
change sign. When such cases arise, the form assumed by the 
curve y —fix) in the immediate vicinity of the point at which 
x — a will be one of those represented 

at A and B in Fig. 13. 

At these points the value of tan </> or 
f\x) is zero, but at A it is positive on 
both sides of the point, and f{x) or y is 
an increasing function, while at B f\x) 
is negative on both sides of the point, Fig. 13. 

and f(x) is a decreasing function. 

136. It is important to notice that at A the value zero 
assumed by f\x) constitutes a minimum value of this function, 
thus a root of f\x) — o for which f\x) is a minimum corre- 
sponds to a case in which f(x) is an increasing function. In 
like manner a root of fix) = o for which f'(x) is a maximum 
is a case in which f(x) is a decreasing function. 

137. It follows from the preceding article and from Art. 
124 that, \if'(a) = o, then, of the two functions f{x) and /'(V), 
one will be a maximum and the other a decreasing function, 
or else one will be a minimum and the other an increasing 
function. Hence, if we consider the case in which the given 
function and several of its successive derivatives vanish for the 
same value of x, it is evident that when these functions are 
arranged in order they will be either alternately maxima and 
decreasing functions, or alternately minima and increasing func- 
tions. 

138. Now suppose that f\x) is the first of these successive 



154 MAXIMA A AW MINIMA. [Art. 1 38. 

derivatives that does not vanish when x — a, then, writing the 
series of functions 

/(*), /'(*), /"(*), /-'(*), f{x), 

let us assume first that /"(a) is positive. Then in the above 
series of functions f*~ z (a), f n ~\d), etc., will be increasing 
functions while f n ~-{a), f n ~\a), etc., will be minima. 

Now whenever n is odd, the original function will belong to 
the first of these classes and will be an increasing function, 
while if n is even the original function will belong to the second 
class and will be a minimum. 

On the other hand, if /"(a) has a negative value, the series 
of functions will be alternately decreasing functions and maxi- 
ma ; and when n is odd fid) will be a decreasing function, but 
when n is even f{a) will be a maximum. 

Thus we shall have neither maxima nor minima unless the 
first derivative, which does not vanish when x — a, is of an 
even order ; but when this is the case we shall have a maximum 
or a minimum according as the value of this derivative is nega- 
tive or positive. 

139. The following function presents a case in which the 
above principle is advantageously employed. 

f{x) = e x + e~ x 4- 2 cos x, 

f\x) = e* — e~ x — 2 sin x. 

Zero is evidently a root of the equation fix) — O.* In this 
case 

* Zero is the only root of f\x) = o in this example ; for 

£ 2x — 2£ x cosx+T (e x — i) 2 
/ (x) = > . 

fix) therefore cannot be negative, hence f'(x) cannot again assume the value 
zero. 



§ XIX.] EMPLOYMENT OF HIGHER DERIVATIVES. J 55 



f'\x) — c- v + c~ x — 2 COS x .'. /"(o) = o, 

f"\x) — c v — c~ x + 2 sin x .'. f'"(o) = o, 

f(x) = e* + e~ x + 2 cos * .-. / iv (o) = 4. 

The fourth derivative being the first that does not vanish, and 
having a positive value, we conclude that x — o gives a mini- 
mum value of f(x). 

Examples XIX. 

1. Show that a£ kx + bs~ kx has a minimum value equal to 2 V(ab). 

Find the maxima and minima of the following functions : 

2. f(x) = x sin #. 

A maximum for a value of x in the second quadrant satisfying the 
equation tan x = — x. 

Z,f{x) =- + 



2 a 

The roots jc = = and x = r give a min. and a max. if b is 

a + a — b ° 

positive, but a max. and min. if b is negative. 

4. /(#) = 2 cos * + sin 2 x. 

Solution : — f \ x ) — 2 sm # (cos x — 1) ; 

rejecting the factor 2(1 — cos x), which is always positive, we put 

v = — sm x. Hence — = — cos x. 
ax 

A max. for x = 2nn ; 

a min. for x = (2;/ -f 1) n. 

5. f{x) = sin x(i + cos #). A max. for x = \n ; 

a min. for jr = — £tt ; 
neither for x — n. 



I5 6 MAXIMA AND MINIMA. [Ex. XIX. 

, rl x sin x cos x 

6. /(*) = 



cos 2 ( \n — x) ' 
Solution : — 
On expanding cos 2 {\7t — x), the reciprocal reduces to 

£ (cot x + 3 tan # + 2 4/3). 

Multiplying the derivative by 4 sin 2 x, we obtain 

7; = 3 tan 2 x — 1 ; 

whence — - = 6 tan x sec 2 .#. 

Therefore tan x = \l\ gives a minimum, and tan x — — \f\ a maximum 
value of the reciprocal. Thus x = \it gives to/(x) a maximum value 
equal to J Y3 and # = — \n, a minimum equal to — 00. 

7. /(.#) = — ; -. A max. when x — cos x. 

1 + x tan x 

/( \ _ ^ 2 + 4 X + IO A min. for # = — 3 ; 

jc 2 + 2^ + 1 1 ' a max. for x — 4. 

9. /(*) = ^ln»«« 8 *. 

Maxima for cos x = ± |/f ; minimum for cos # = o (the angles 
being taken in the fir|t semicircle). 

10. f{x) = sec x + log cos 2 .*. 
Multiplying the derivative by cos 2 .*, we obtain 

v = sin Jt(i — 2 cosjc). 

A max. for .# = o, and x — n ; 
a min. for # == ± \ n. 



tan 3 # A min. for x — o, §7T, f 7r, and ?r ; 

tan $x ' a max. for x = J-7T, -J-^r, ^-zr, etc. 



1 2 . /(#) = e x -\- a x — x*. A min. for # = o. 



§ XIX.] EXAMPLES. 157 

13. /(a) — 4.V- -f cos 2x — I (t 2 * + £ -2j ). A max. for x = o. 

14. f(x) = (3 — a?)fc ta — 4JC£" — a. Is there a maximum or a mini- 
mum corresponding to x = o ? Neither. 



XX. 

Implicit Functions. 

140. Let the relation between the variables ;r and j/ be ex- 
pressed by the equation 

F(x,y) =0; . (1) 

and let it be required to find the maxima and minima values 

of either variable. The value of the derivative -^- may be 

ax 

found as in Art. 66, in the form 

f = ^ (2) 

ax v 

in which u and v are in general functions of x and y. 

If equation (1) be regarded as the equation of a curve, it is 
evident that wherever the tangent to the curve is parallel to 
the axis of x we shall have 

u = o (3) 

Hence the values of x and y which satisfy simultaneously 
equations (1) and (3), and do not make v = o, will give points at 
which the ordinate is a maximum or a minimum according as 
the value of the second derivative is negative or positive. The 
corresponding value of the second derivative is most readily 
obtained bv the formula deduced in the next article. 



158 MAXIMA AND MINIMA. [Art. 141. 

141. From equation (2) we obtain 

du dv 
d y dx dx , N 

1* = v' (4) 

In the case which now presents itself we have u = o, which 

causes the second term of the numerator to vanish, and more- 

dv du 

over, since -f- = o, the value of — is that which would be ob- 
dx dx 

tained by taking the derivative of u on the supposition that y 

is constant. Hence, indicating by brackets the particular 

values which the derivatives take when -f- — o, we have 

dx 



c 



^~du 

dy-i_ldx~_ 

dx'A v 



(5) 



- in which the values of x and y determined by equations (1) and 
(3) are to be substituted. 

142. To illustrate, let 

xy 1 — x^y = 2a\ (1) 

in which a denotes a positive constant. Differentiating 

2 . dy „ dy 

y + 2xy -i- — 2xy — x 1 -^- — o ; 
dx dx 

therefore, ± = y _i^-y) 

dx x[2y — x) 

In this example u = y (2x — y) and v — x(2y — x) ; putting 
u = o, we obtain 

y = o or y = 2x. 



§ XX.] 



IMPLICIT FUNCTIONS. 



I 59 



Substituting y—o in (1) gives an infinite value to x, but 
y — 2x gives 

x — a and y = 2a. 

Formula (5) of the preceding article gives 



~dy 



] 



2y 



_dx- J x\2y — x) 
substituting x — a and y — 2a, we find 



.ax I a, 2a 



3" 



which is positive ; hence 2a is a minimum value of y. 

In like manner maxima and minima values of x may be 
found by employing the condition 



v = o, 



and a formula, similar to (5) of Art. 141, to discriminate between 
the maxima and minima. 



143. Fig. 14 represents the curve corre- 
sponding to equation (1) of the preceding 
article; viz., 



xy 1 



x*y = 2a z . 



(1) 




The ordinate at A is the minimum found 

above, and the abscissa of B (—2a, — a) is / 

a maximum. Fig. 14. 



A / 



Infinite Vahces of the Derivative. 
144. If y denotes a function of x, a maximum or a minimum 

of y occurs whenever the derivative -f- changes sign as x 

ax 



160 MAXIMA AND MINIMA. [Art. 1 44. 

passes through a certain value. Since a function may change 
sign on passing through infinity, it is necessary to consider the 
cases in which 

dy 



But, since in such cases 



dx 



dx _ 

dy 



this condition usually corresponds to a point like B in Fig. 14 
at which x, instead of y, is a maximum or a minimum. In 
such cases x does not pass through the value which makes the 
derivative infinite, and therefore the condition necessary for a 
maximum or minimum value of y is not fulfilled. The function 

y — (x — a) 1 

presents a case of this kind ; for 

dy I 



dx 4(> — dy ' 

which is infinite for x = a, but, since y becomes imaginary, x 
cannot pass through the value a ; an essential condition for a 
maximum or a minimum value of y is therefore wanting, but it 
is easily shown that a minimum abscissa of the curve j/ 4 = x — a 
occurs at the point in question {a, o). 

145. It is noteworthy also that since by Art. 104 f'(x) is 
infinite whenever fix) is infinite for a finite value of x y the 
roots of the equation 

/'« = * 

will include the values of x which render f{x) infinite when- 
ever such values exist. Thus, if 

-. N <£X 



I XX.] 



INFINITE VALUES OF THE DERIVATIVE. 



161 



/"(*) = - 



a\x + a) 



x = a makes f\x) infinite, but it also makes fix) infinite.* 

Putting f (x) = o, we have x = — a, which corresponds to 
a minimum value of f{x), since, for this value of x, f\x) is an 



increasing function. 



146. If the equation 



/'(*) 



has a root, for which f{x) remains finite and does not be- 
come imaginary, it is still necessary to ascertain whether f\x) 
changes sign, since there will be a maximum if the change be 
from + to — , and a minimum if it be from — to +. 
The form of the curve 

in the vicinity of a maximum or a minimum ordinate of this 
variety is represented at A and B in Fig. 15. 
As an example, let 



whence 



f\x) = \x-\x^ - «*)"*. 



k 



f\x) is infinite when x= o and when x — b. 
When x — o f\x) does not change sign, 
since x~* cannot be negative, but when x — b 
it changes sign from — to -f ; hence f(x) has 
a minimum value when x — b. 



Y 



Fig. 15. 



* When as in this example /"'(.*•) changes sign as x passes through the value that 
makes f(x) infinite, this value of f(x) may be regarded as an infinite maximum. 
See the corresponding curve Fig. 17, Art. 202. 



1 62 MAXIMA AND MINIMA. [Ex. XX. 



Examples XX. 

i. Given 25/* — 6xy + # 2 — 9 = o, to find the maxima and min- 
ima of y. Min. for x = — f ; max. for x = j- . 

2. Given ^ 4 + 2ax*y — ay 3 = o, to find the maxima and minima 
of y. Min. for x = ± # . 

3. Given jjr — jc 2 _y + x — x* = o, to prove that .# = — 1 gives a 
maximum value of y. 

4. Given sa^y^-h xy % + 4^ = o. Show that when # = \a, y has 



a maximum value, namely — 30, the value of 
_8_ 
5* ' 



djr 



being then 



5. Given y 4- 2# 2 _y + \x — 3, to prove that x = 1 gives a maxi- 
mum value of jy. 

6. Given j 3 + x % — $axy = o. A max. for .%• = a V2; 

a min. for x — o. 

7. Find maxima and minima of the following functions : 

f{x) = (#* — b*Y, A min. for x = o. 

8. f(x) = (x* — £ 9 )$, A max. for ^ = o; 

a min. for :r = ± <£. 

9. /(.#) = (^ 2 4- sx + 2) 5 + # 5 . 

f(x)= 00 gives min. corresponding to # = — 2, x= — 1 and^ = o. 
f'(x) = o gives two intermediate maxima. 

10. f{x) = (V 4- 2*)^— (# + 3)3. Max. for * = J(— 3 ± 4/17); 

min. for # = o and # — — 2. 

n. /(#) = 7 -r- 9 . See Art ia<. A min. for x = o. 

y V 7 (X + 2) 2 ^° 



§ XX J EXAMPLES. 163 

12. f(x) = (x — a)> (x — b)* + c. A max. for x — ; 

3 



min. for x = a and x = b. 
3- /(- 



/-/ \ (x — a)(x — b) a • r 2 ^ 

13- /(- v ) = * —* ' • A mm. for x = 



a + b 

14. f(x) = (x — a)*(x — b)». 

Solutions for x — a and jc = J (2^ + «) ; if £ > 0, the former gives 
a max. and the latter a min. 

15. f(x) — log cos x — cos*. Max. for x = o. 

16. /(*) = (* - i) 3 (* + i)- 2 . Max. for x = — 5. 



Miscellaneous Examples. 

^c — 1 

1. /(je) = -= = . Use the reciprocal. 

v x — 3.*- H- 2jc + 54 

Max. for x — 4, 

2. /(*) == — ; . A max. for x = o; 

x . + x ~~ * a m in. for .t = 2. 



3. /(.v) = x a e f ". A min. for x = — . 

4. The equation of the path of a projectile being 

* X s 

y = .r tan a — 



4// cos a 

find the value of jf when y is a maximum ; also the maximum value 
of y. Max. when jc = h sin 2 or, and y = /* sin 2 or. 

5. In a given sphere inscribe the greatest rectangular parallel- 
opiped. 



164 MAXIMA AND MINIMA. [Ex. XX. 

Solution : — 

Regarding any one edge as of fixed length, it is easy to show that 
the other two edges are equal. Hence the three edges are equal. 

6. In a given cone inscribe the greatest rectangular parallelo- 
piped. 

Solution : — 

Regarding the parallelopiped as inscribed in a cylinder which is 
itself inscribed in the cone, the base is evidently a square, and the 
altitude is that of the maximum cylinder. See Ex. XVII, 9. 

7. Given u = a cos 2 x + b cos 2 j, (1) 

x and y being connected by the equation 

y-x = i7t; . . . . . . . . (2) 

find the values of x when u is a maximum or a minimum. 

Solution : — 

die . _ . dy 

— - = — a sin 2x — sin 2y — . 
dx dx 

From the second equation we have 

-j- = 1, and 27 = 2X + \n ; 
dx 

whence sin 27 = cos 2x ; 

therefore -7- = — a sin 2x — b cos 2x = 0, 

dx 

and jc = J tan _1 ( 



This equation has a root in each quadrant, and, as u does not admit of 
infinite values, these roots correspond to alternate maxima and minima. 

Since — = — b, it is a decreasing function when x = o ; therefore 

dxjo 

the value in the first quadrant gives a minimum. 



^ XX.] MISCELLANEOUS EXAMPLES. 165 

8. Given one angle A of a right spherical triangle, to find when the 
difference between the sides which contain it is a maximum. 

We have in this case tan c cos A = tan b, in which b and c are the 

variables and. since c — b is a maximum. - = 1. 

do 

A max. when b — tan _1 ( Vcos A). 

9. A Norman window consists of a rectangle surmounted by a 
semicircle. Given the perimeter, required the height and breadth of 
the window when the quantity of light admitted is a maximum. 

The radius of the semicircle is equal to the height of the rectangle. 

10. A tinsmith was ordered to make an open cylindrical vessel of 
given volume, which should be as light as possible ; find the ratio be- 
tween the height and the radius of the base. 

The height equals the radius of the base. 

n. What should be the ratio between the diameter of the base and 
the height of cylindrical fruit-cans in order that the amount of tin used 
in constructing them may be the least possible ? 

The height should equal the diameter of the base. 

12. Determine the circle having its centre on the circumference of 
a given circle so that the arc included in the given circle shall be a 
maximum. 

A max. for the value of which is in the first quadrant. 

13. Given the vertical angle of a triangle and its area ; find when 
its base is a minimum. The triangle is isosceles. 

14. Prove that, of all circular sectors of the same perimeter, the 
sector of greatest area is that in which the circular arc is double the 
radius. 

15. Find the minimum isosceles triangle circumscribed about a par- 
abolic segment. 

The altitude of the triangle is four-thirds the altitude of the seg- 
ment. 



1 66 MAXIMA AND MINIMA. [Ex. XX. 

1 6. Find the least isosceles triangle that can be described about a 
given ellipse, having its base parallel to the major axis. 

The height is three times the minor semi-axis. 

17. Inscribe the greatest parabolic segment in a given isosceles 
triangle. 

The altitude of the segment is three-fourths that of the triangle. 

18. A steamer whose speed is 8 knots per hour and course due north 
sights another steamer directly ahead, whose speed is 10 knots, and 
whose course is due west. What must be the course of the first steamer 
to cross the track of the second at the least possible distance from her ? 

N. 53° 8' W. 

19. Determine the angle which a rudder makes with the keel of a 
ship when its turning effect is the greatest possible. 

Solution : — 

Let $ denote the angle between the rudder and the prolongation 
of the keel of the ship ; then if b is the area of the rudder that of the 
stream of water intercepted will be b sin (f> : the resulting force being 
decomposed, the component perpendicular to the rudder contains the 
factor sin 2 <j>. Again decomposing this force, and taking the compo- « 
nent that is perpendicular to the keel of the ship, which is the only- 
part of the original force that is effective in turning the ship, the ex- 
pression to be made a maximum is 

sin 2 (f> cos (f>. 
Whence we obtain 

tan (f> = 4/2. 

20. The work of driving a steamer through the water being propor- 
tional to the cube of her speed, find her most economical rate per hour 
against a current running a knots per hour. 

Solution : — 

Let v denote the speed of the steamer in knots per hour. The 
work per hour will then be denoted by kv z , k being a constant, and the 
actual distance the steamer advances per hour by v — a. The work 
per knot made good is therefore expressed by 

v — a ' 



§ XX.] MISCELLANEOUS EXAMPLES. 1 67 

Whence Ave obtain the result 

v = \a. 

21. To find the parallel on the earth's surface at which the difference 
between the geographical and the geocentric latitude is greatest. 
Solution : — 

Assuming the meridian to be an ellipse, and taking the origin at the 
centre, its equation will be 

Let B denote the geocentric latitude, and ip the geographical lati- 
tude ; then 

. , v tan # — tan e 

tan (?/.' — 0) 



1 + tan ip tan 
is to be a maximum, tan Q = - , and, since ip = (f> — %7t y 



dx 

tan ib = — cot <p — — . 

dy 

Substituting these values in the expression for tan (ip — 6) we obtain 
x = — which gives the maximum required. 

22. The position of a point in each of two media (A and B) sepa- 
rated by a plane surface, being given, it is required to find the path 
described by a particle which goes from one point to the other in the 
shortest possible time, the velocity of the particle being constant in 
each medium. 

Solution : — 

The path is evidently composed of two straight lines, one in each 
medium. It is easily proved that the plane passing through the two 
points, and normal to the separating surface, contains these lines. For, 
if not, let the path be projected on this plane, then the portion of the 



l68 MAXIMA AND MINIMA. [Ex. XX. 

new path thus formed, contained in each medium, will be shorter than 
the corresponding portion of the original path ; therefore the new 
path will be described in less time than the original path. 

Let a and b denote the lengths of the perpendiculars let fall from 
the given points upon the separating surface, u the velocity in the 
medium A, and v that in the medium B. 

Let the particle be supposed to move from A to B, and let i denote 
the angle of incidence, that is, the angle between a and the path in A, 
and r the corresponding angle in the medium B, 

If c denotes the projection of the path upon the separating surface, 
which is constant, the relation between i and r is expressed by 

a tan i + b tan r = <r (i) 

The time occupied in describing the path is 

a sec i b sec r , » 

1 ; (2) 

u v 

putting the differential of this expression equal to zero, we have 

a .... b . . 

— sqc 1 tan i at = seer tan r dr (v 

From (1) we obtain 

a sec 2 i di— — b sec 2 r dr ; (4) 

dividing (3) by (4) to eliminate di and dr, we have 

i_ tan / _ 1 tanr 
u sec i v sec r ' 

v sin r 

or — = . 

u sin/ 

The resulting path is that actually described by a refracted ray of 
light, if we suppose the ratio of the velocities to equal the index of re- 
fraction. 

This problem was originally proposed by Fermat. 



§ XX.] MISCELLANEOUS EXAMPLES. 1 69 

23. The orbits of Venus and the Earth being regarded as circular, 
determine the distance between the planets when the brightness of 
Venus is a maximum. 

Solution : — 

Let r denote the required distance, a the radius of the Earth's orbit, 
b the radius of the orbit of Venus, and (j) the angle between b and r. 

Denoting the apparent semi-diameter of the disk at the distance 

unity by c, at the distance r it will be denoted by - , the correspond- 
ing area of the entire disk by — j , and the illuminated portion of the 
disk by 



We therefore require the maximum value of 











I - 


f cos <p 










r~ 




hence, 


eliminating 


cos 


by 


the relation 












a- = 


lr + 


r" — 2br 


cos 


A 


we obtain 




f(r)- 


_b 2 


— a' + r 2 

a 


+ 


2b r 

t 



(I) 

and, solving the equation f\r) = o, we derive 

r — V{b' + 3<r) — 2b, (2) 

the other root being inadmissible since it is always negative. 

If we regard the brightness as a function of the time, the equation 
for determining the maxima and minima oif(r) is 

Hence the roots of the equation 



dr 

It 



I70 MAXIMA AND MINIMA. [Ex. XX. 

are also solutions of the problem. These roots evidently correspond 
to the maxima and minima values of r, which occur at the conjunctions ; 
at each of which the brightness is obviously a minimum. Hence at 
the points determined by equation (2) it is a maximum. 

Taking the mean distance a of the Earth from the sun as unity, 
that of Venus according to G. W. Hill is 0.723 ; substituting these 
values for a and b in the above expression for r, we obtain r = 0.431, 
the corresponding elongation (angle between a and r) will be found to 
be 39 43'. 



CHAPTER VII. 

The Development of Functions in Series. 



XXI. 

The Nature of an Infinite Series. 

147. A FUNCTION which can be expressed by means of a 
limited number of integral terms, involving powers of the inde- 
pendent variable with positive integral exponents only, is called 
a rational integral function. 

When fix) is not a rational integral function, it is usually 
possible to derive an unlimited series of terms rational and in- 
tegral with respect to x, which may be regarded as an algebraic 
equivalent for the function. The process of deriving this series 
is called the development of the function into an infinite series. 

When the given function is in the form of a rational frac- 
tion, the ordinary process of division (the dividend and divisor 
being arranged according to ascending powers of x) suffices to 
effect the development. Thus — 

I + 2X + 2.T 2 + 2X* + ' • • , 



— X 



a series of terms arranged according to ascending powers of x, 
each coefficient after the absolute term being 2. 

It is to be observed, in the first place, that, owing to the 
indefinite number of terms in the second member, the equa- 
tion as written above cannot be verified numerically for an 
assumed value of x. In this case, however, the process not 



172" THE DEVELOPMENT OF FUNCTIONS. [Art. 1 47. 

only gives us the series, but the remainder after any number of 
terms. Thus carrying the quotient to the term containing x tl , 
and writing the remainder, we have 

1 A- x 2x nJr x 

— - — = I + 2X + 2X 2 • • • -f 2X n H . 

I — X I — X 

This equation may now be verified numerically for any assumed 
value of x ; or algebraically by multiplying each member by 
1 — x, thus obtaining an identity. 

The ordinary process of extracting the square root of a 
polynomial furnishes an example of a series which may be ex- 
tended so as to include as many terms as we please ; but this 
process gives us no expression for the remainder 

148. Assuming that f(x) admits of development into a 
series involving ascending powers of x, and denoting the re- 
mainder after n + 1 terms by R f we may write 

f{x) = A + Bx + Cx'A- - • • + Nx n + R, . . . (1) 

in which A, B, C, . . . N denote coefficients independent of x> 
and as yet unknown ; the value of R is however not indepen- 
dent of x. If the coefficients B, C, . . . N admit of finite 
values, it may be assumed that R is a function of x which van- 
ishes when x = o ; and in accordance with this assumption 
equation (1) becomes, when x = o, 

f(o) = A, . (2) 

which determines the first term of the series. If in any case 
the value of /(o) is found to be infinite, we infer that the pro- 
posed development is impossible. 

149. When the coefficients B y C, . . . N admit of finite 
values, and the value of the function to be developed remains 



§ XXL] INFINITE SERIES. 1 73 



finite, R will have a finite value. If moreover the value of R 
decreases as n increases, and can be made as small as we please, 
by sufficiently increasing ;/, the series is said to be convergent, 
and may be employed in finding an approximate value of the 
function f(x) ; the closeness of the approximation increasing 
with the number of terms used. A series in which R does not 
decrease as ;/ increases is said to be divergent. 

When the successive terms of a series decrease it does not 
necessarily follow that the series is convergent ; for the value 
of the equivalent function, and consequently that of R, may be 
infinite. To illustrate, if we put x — I in the series 

x + \x* + \x* ! + \x* + • • ■ , 
we obtain the numerical series 

* + i+3 1 + i-r-i+- • • ; 

it can be shown that, by taking a sufficient number of terms, the 
sum of this series may be made to exceed any finite limit, the 
value of the equivalent or generating function of the above series 
being in fact infinite when x = I .* 

150. Since R vanishes with x, every series for which finite 
coefficients can be determined is convergent for certain small 
values of x. In some cases there are limiting values of x, both 
positive and negative, within which the series is convergent, 
while for values of x without these limits the series is diver- 
gent. These values of x are called the limits of convergence. 

* If we consider the first two terms separately, and regard the other terms as 
arranged in groups of two, four, eight, sixteen, etc., the groups will end with the 
terms \, }, -j 1 ^, ^ 2 "» etc. The sum of the fractions in the first group exceeds \ or \, 
the sum of those in the second exceeds f or \, and so on ; hence the sum of 2N such 
groups exceeds the number N, and N may be taken as large as we choose. 

The generating function in this case is log , and unity is the limit of con- 



174 THE DEVELOPMENT OF FUNCTIONS. [Art. 150. 

We shall now demonstrate a theorem by which a function 
in the form f(x Q + ft) may be developed into a series involving 
powers of h, and in Section XXII we shall show how this 
theorem is transformed so as to give the expansion of f(x) in 
powers of x. 



Taylor s Theorem. 

151. A function of h of the form f{x Q + ft) in general admits 
of development in a series involving ascending powers of ft. 
We therefore assume 

f{x Q + ft) = A Q + BJi -f C Q ft 2 + • • • + N h n + R Q , . . (1) 

in which A Q , B Q , Co, . . . N Q are independent of k, while R 
is a function of /z which vanishes when ft is zero. Hence, mak- 
ing ft = o, we have 

f(x ) = A Q . 

We have now to find the values of B , C Q , . . . iV Q , which 
are evidently functions of x Q . For this purpose we put 

x z — x -f /*> whence // = ^ — jt ; 

substituting, equation (1) takes the form 

f( Xi )=/( Xo ) + B (x I -x )+C (x 1 -x o y ■ • ■ +N (x I -x ) H +Ro, 

in which we may regard x z as constant and x Q as variable. Re- 
placing the latter by x, and its functions, B 0} C Q , . . . -Ao, and 
i?o, by i?, C, . . . A 7 , and R, we have 

/(>,) = f{x) +B{x, -x) + C{x x - xf • • • + N(x z - *)" + £. . (2) 

Taking derivatives with respect to x, we have 



§ XXL] TA YLORS THEOREM. 175 

O = /' (x) -B + (.v, - .r) g - 2C(.v, - x) + (x, - *)' g ' • ■ 

-, l N(x t -xy-< + (x t -xy^ + ^. ... ( 3 ) 

To render the development possible, B, C, . . . N, and R must 
have such values as will make equation (3) identical, that is, true 
for all values of x. 

152. It is evident that B may be so taken as to cause the 
first two terms of equation (3) to vanish, and that, this being 
done, C can be so determined as to cause the coefficient of 
(jfj — x) to vanish, D so as to make the coefficient of (x x — xf 
vanish, and so on. The requisite conditions are 

r, t \ d dB r* dC ^ 

f (x) - B = o, ~-2C= o, — - iD = o, etc., 

A £ 11 ' \» dN . dR « 

and finally (x z — x) — — h -=- = o. 

From these conditions we derive 

B=f{x), C = i^=i/"(x), 

D = *fx = Tk f " {x) ' E =T^r A ™> 



and in general JSf = f n (x\ 

& i-2 • • >n J v } 

Putting x for x, and substituting in equation (1) the values of 
A , B , Co, . . . N , we obtain 

f{x + k)=f(x ) +f( Xo )h+f\x ) -*1 ■ • . +/» (*„) A — - +i? . (4) 

1 * £ 1 *2 • • • fl 



I76 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 52. 

This result is called Taylor's Theorem, from the name of its dis- 
coverer, Dr. Brook Taylor, who first published it in 171 5. 

It is evident from equation (4) that the proposed expansion is 
impossible when the given function or any of its derived func- 
tions is infinite for the value x Q . 



Lagrange s Expression for the Remainder, 

153. R denotes a function of x which takes the value R 
when x = x Q , and becomes zero when x — x T . It has been 
shown in the preceding article that R must also satisfy the 
equation 

, , n dN dR 

or, substituting the value of N determined above, 
dR _ (x T — x) n 



dx 1 • 2 • • • 11 



/" +, W (5) 



This equation shows that — cannot become infinite for any 

value of x between x Q and x If provided f" + I (x) remains finite 
and real while x varies between these limits. Since it follows 
from the theorem proved in Art. 104 that all preceding deriva- 
tives must be likewise finite, the above hypothesis is equivalent 
to the assumption that/(;r) audits successive derivatives to the 
(n + i)th inclusive remain finite and real while x varies from x 
to x + h. 

154. Let P denote any assumed function of x which, like 
R, takes the value R Q when x = x Q and the value zero when 

dP 

x = x z ,3ind whose derivative - r - does not become infinite or 

dx 

imaginary for any value of x between these limits. 



§ XXL] EXPRESSIONS FOR THE REMAINDER. 177 



Then, R being assumed to be finite, P — R denotes a func- 
tion of x which vanishes both when x — x Q and when x — x t 
and whose derivative cannot become infinite for any interme- 
diate value of x. It follows therefore that the value of this 
function cannot become infinite for any intermediate value of x. 

Since, as x varies from x to x iy P — R starts from the value 
zero and returns to zero again, without passing through infinity, 
its numerical value must pass through a maximum ; hence its 
derivative cannot retain the same sign throughout, and as it can- 
not become infinite it must necessarily become zero for some 
intermediate value of x. Since x x =x Q + Ji this intermediate 
value of x can be expressed by x Q + d/i, 6 being a positive proper 
fraction. It is therefore evident that at least one value of x 
of the form 

x = x + Bh 

will satisfy the equation 

dP dR ,~ 

*S=° (6) 

155. The value of P will fulfil the required conditions if we 
assume 

k n+1 

for this function takes the value R Q when x = x Q and vanishes 
when x = x x ; moreover its derivative with reference to x, viz., 

M ^'-^.A, .... (7) 



dx k 



W-j-I 



does not become infinite for any intermediate value of x. Sub- 
stituting in equation (6) the values of the derivatives given in 
equations (5) and (7), and solving for R Q , we obtain 

^0= — \— — .f H+ \x + d/i). . . . (8) 

1-2 • • • « • (11 4- 1) 



178 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 5 5. 

This expression for the remainder was first given by La- 
grange. 

The series may now be written thus : 

/(*,+ h) =/(*o) +f'(*o)k + /"{*,) ^ ■ ' - 

+ f"(*o) + f n +\X o +0k) ■ ^ ¥ ) r . . . . ( 9 ) 

It should be noticed that the above expression for the remain- 
der after n + 1 terms differs from the next, or (n + 2)th term 
of the series, simply by the addition of 6/1 to x Q . 



A Second Expression for the Remainder. 

156. The function P will also fulfil the conditions specified 
in Art. 154 when assumed in the form * 

p- r /'(*■)-/"(*) 

since this function becomes equal to R when x = x oy and is 
zero when x = x 1 ; and moreover its derivative ; viz., 



^ /-(*.)-/"(*.)' 



* The form 

yfo) - <?(*) 

in which q> is continuous between q>{xi) and <p(x ), includes all the forms in which 
P can be assumed. The resulting value of P is 

which is accordingly the most general form of R that can be derived by this method, 
and includes that of Cauchy as well as other forms of the remainder. See The 
Mathematical Messenger for April, 1873, in which this method of proving Taylor's 
Theorem was first published. 



§ XXL] EXPRESSIONS FOR THE REMAINDER. 1 79 

is continuous between the limits x = x Q and x — x v , because 
f' ! '\x) is, by the hypothesis of Art. 153, continuous between 
these limits. 

Employing this expression for -7- in place of that given in 

equation (7), we obtain 

R Q _ (x,- x Q - e/i) n = h\\ - 8) n 

f n {*x) -f"{xo) 1-2 ■ • • n i-2 • • • n ' 

Since (1 — 6) H is a positive proper fraction, it may for simplicity 
be denoted by 6 ; hence we have 

Ro=eif%x + k)-f n {x y\—-^- — . . . (10) 

Since this value of R falls between 

o and [/"(*„+ A) -/><,)] — -— , 

1-2 • • • n 

the value of f{x + h) is intermediate between the two expres- 
sions, 

h n 



f(x )+f(x )k- • >+/*(*) 



and f(x ) + f\x )h • • • + f n (x + h) 



1.2 • . • n 
ti l 



1-2 ... n 



It is to be noticed that the latter differs from the former only 
in the substitution of x Q + h for x Q in the last term. 



Limits to the Application of Taylor s Theorem. 

157. When, for a given value of h, the hypothesis stated in 
Art. 153 is true for every value of n, the expressions for R OJ and 



80 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 57. 



consequently the complete form of Taylor's Theorem expressed 
in equation (9), Art. 155, are applicable for all values of n. 

In many cases, however, when the value of x is given, there 
exist limiting values of h, either positive or negative, such that 
for numerically greater values of h equation (9) is not true when 
n is unrestricted. These values of h are of course the numer- 
ically least positive and negative values of h that cause one of 
the derivatives to become infinite when we put x = x Q -\-h. 

Equation (9) is true even when h is taken equal to the limit- 
ing value ; for, since the hypothesis of Art. 153 only requires 
that the derivatives shall not become infinite for values of x be- 
tween x and x + It, this case is not excluded. [See examples 
8 and 9, below.] If, however, it is the function itself (not merely 
one of its derivatives) that becomes infinite, equation (9) is in- 
applicable because R Q is likewise infinite. [See Art. 154.] 

When h is taken greater than the limiting value, equation (9) 
still holds true when n + I is less than the index of the first 
derivative that becomes infinite. [See example 10, below.] 

158. In many cases the limit of convergence is numerically 
less than the limiting value of h mentioned in the preceding 
article, which refers only to the applicability of the expression 
iox R ; since, for certain values of h, although the expression 
for R Q is applicable, its value may fail to decrease as n increases. 
The existence of this case is generally indicated by an increase 
in the value of the successive terms of the series. See Art. 160. 
Example 7, below, illustrates the applicability of equation (9) 
to a case in which h exceeds the limit of convergence. 



The Binomial Theorem, 

159. We shall now apply Taylor's Theorem to the function 
(a -f b) m in order to obtain a series involving ascending powers 
oib. 



§ XXL] THE BINOMIAL THEOREM. l8l 

In this case b takes the place of h, and a that of ,i' ; hence 

(jr)=«Mjr .'. f\Xo) — mx —ma 

f"(x)—m(m — i)x m ~ l .'. f"(x ) = w(7/i — i)xo~~ — in{m— i)d"~ 
and 

f"{x^ = m(m — i)(;« — 2) • • • (;« — n + i)tf'"~*. 

Whence 

(tf + 0) = tf + ;;z# # H a o • • • • 



;«(;// - i)(ot - 2) • • . (;;/ -11 + i ) jOT .„^ + .-. . 
I.2.3 • ' • * 

This result is called the Binomial Theorem. 
Each term of the series can be derived from the preceding 
term by multiplying by a factor of the form 



m — 11+1 b 



m + 1 



a L n 



If m is a positive integer, this factor will become zero for 
n — vi + 1. When this is the case, the series will consist of a 
finite number of terms ; otherwise, as n increases, the quantity 
in brackets will approach indefinitely to — 1, and the terms 
will ultimately decrease numerically if b is less than a, but they 
will ultimately increase numerically if b exceeds a. 

160. Assuming a positive, if b is likewise positive, the hypo- 
thesis of Art. 153 holds for all values of n\ but it does not 
hold when b is negative and numerically greater than a, 
since b — — a makes f'\a + b) — 00 when n > m. Hence for 
all values of b algebraically greater than — a we are entitled 



1 82 THE DEVELOPMENT OF FUNCTIONS. [Art. l6o. 

to use Lagrange's expression for the remainder, which gives in 
this case 

= m{m-i){m-2). . . {m - » ) ^y,.,,-,^ . 

1-2 • • • (n + i) v J 

The ratio of the remainder to the first term omitted is 



(a + db) m - n ~ x _ / a V 
0**-"- 1 " \a + db) 



The value of the fraction 7-7 is always between unity and 

a + 6b J J 

; hence, when the terms of the series decrease in value 



a + b 

indefinitely, so likewise will the successive values of the re- 
mainder; that is, the series will be convergent. Therefore the 
limit of convergence for positive values of b is b = a. See 
Art. 158. 

Examples XXI. 

1. To expand log (x + H) by Taylor's Theorem. 
Solution : — 

/(* +^) = log(* +>S) 

f(x) = \ogx .'. f(x ) = \ogx 

Ji. ^Q 



/'"(*) = £ ••• /'"(*.) = ^ 



8 xxi.] 



EXAMPLES. 



83 



/*(*) 



'• /*(*<,) = - 



1-2-3 



r{x )=- { -,y Li^zl) : . r{Xo) = _ ( _ 1} . i-» (- 



a:! 



By substituting in equation (4), Art. 152, we obtain 



log(* +£) = log* + -- — + r ... -(-jy + j r 

x o 2x- Q $x A 4 x 4 nxl 

Employing Lagrange's expression for the remainder (Art. 155) we 
derive 

7-n4-l 



(n + i)(* + S/O"" 1 



2. Expand a Xo+h . 
Solution : — 

f(x Q + h) = of° + h 

f(x) = a 1 :. f{x Q ) = a*° 

fix) — log a-a* .'. f\x Q ) = log a-a x ° 

f"(x) = (log of- a? ;. f"( Xo ) = (log ay.a*° 

f\x) = (log a):<f .'. f n {x Q ) = (log «)•.<**. 
Substituting in equation (4), Art. 152, we have 

*•+» = *. f, + log ,.* + (log .)■•£..-..+ (los " ) "" // 



1-2 



I • 2 • • • 11 



+ R~ 



Equation (10), Art. 156, gives 

i? =e(iogtf)"^°[> ft -i] 



1 • 2 • • • 11 



1 84 THE DEVELOPMENT OF FUNCTIONS. [Ex. XXL 

3. Find the expansion of f(x Q + h), when f{x) — x log x — x, writ- 
ing the (n + i) th term of the series. 

f(x Q + h) — x Q \ogx —x Q + \ogx -/i + + ... 

X Q 1-2 Xl 2-3 

+ ( ~ l),l ^r r '(^-i)^'" 

4. Expand sin -1 (x Q + ^) to the fourth term inclusive. 
sin _1 (^ + h) — sin _1 ^ H - r + ^ — T • 

(1-*!)* (i-^) f M 

1 -f 2x'i K a 



+ 



sm 



5. Prove that 
z \_ 1-2 1-2-3 I-2-3-4 

I-2-3-4-5 J 

6. Prove that 

tan (Jtt + ^) = 1 + 2h + 2^ + -p 3 + W + • • ' 

7. Find the value of the proper fraction which satisfies equation 
(9), Art. 155, when f(x) = x z , assuming x Q = i, h = 3, and n= 2 ; 
also find for the same values of x and /z, when ?z = 3. 

For « = 2, 6 = ; for « = 3, = 0-15 nearly. 

337° 8 

8. Expand (— 1 + ^) 3 , and show that the limit of applicability of 
equation (9), Art. 155, [see Art. 157] is h = 1 : find also, for this value 
of h y the value of when ;z = 2, and when « . = 3. 

For # = 2, =1 — ^\ 4/7 ; for n = 3, = 1 — (jV) b - 

9. Given /(#) = # log,* — x t show that the limit of applicability of 
equation (9), Art. 155, to/(i + A), for negative values of A, is/£ = — 1, 



§ XXL] EXAMPLES. 185 



and determine the values of for this value of h, when n = 1, and 
when // = 2. (See example 3.) 

For n = 1, = J ; for n = 2, 9 = 1 — \ V3. 

10. In the case of the function (— 1 + h)\ show that if h = 2, 
equation (9), Art. 155, is not necessarily true when u exceeds unity, 
and determine the value of corresponding to n = 1. = Jff. 

11. By putting # = 1 and ^ = 1 in the series obtained in example 
3, derive a numerical series for computing 2 log 2, and, making n — 6, 
show by the method of Art. 156 that 2 log 2 is between 1.400 and 
1.368. 



XXII. 

Maclaurin s Theorem. 

161. We shall now give a particular form of Taylor's Series, 
which is usually more convenient, when numerical results are to 
be obtained, than the general form given in the preceding sec- 
tion. 

This form of the series is obtained by putting x Q = o and 
replacing h by x in equation (4), Art. 152. Thus, 

/(*)=/(o)+/'(o)*+/"(o) g. • ■ + /»(o) j^-^ + R„ . . (1) 

and, the same substitutions being made in equation (8), Art. 155, 
we obtain 

R =r+\6x)-- 



...(»+ i) 

In like manner, by means of the result obtained in Art. 156, it 
may be shown that the value of f(x) is between that of the 
two expressions, 



1 86 THE DEVELOPMENT OF FUNCTIONS. [Arjt. l6l. 

Ao)+fio)x.--+f\o) i *" t , 



and /(o)+/'(o>- • •+/» 



1-2 • 



When ./"(■*) is denoted by 7, equation (1) is written thus: — 



x 2 



162. The series given in equations (1) and (2), although first 
discovered by Sterling, has received the name of Maclaurin's 
Series. It was published by Maclaurin in 1742; but it does 
not appear that he ever intended to claim it as his own dis- 
covery. 

As a mode of development Maclaurin's Theorem is in reality 
no less general than that of Taylor ; for any function which is 
included in the general form f(x Q + h) may also, by giving a 
different signification to the symbol /, be expressed in the 
form fill), or, employing the notation of the last article, fix). 
Thus, if log (1 + h) is to be developed by Taylor's Theorem, 
fix)— log x, the value of x Q being unity; but, if log (1 + x) 
is to be developed by Maclaurin's Theorem, we must put 
f(x) = log(i + x). (Compare Ex. XXL, 1, with Art. 164.) 



The Exponential Series and the Value of s. 

163. As an example of the application of the above theo- 
rem, we shall deduce the development of the function e x , which 
is called the exponential series, and shall thence obtain a series 
for computing the value of £. 

The successive derivatives of z x being equal to the original 
function, the coefficients, /(o), f'(6), etc., each reduce to unity; 



§ XXII.] THE EXPONENTIAL SERIES. 1 87 

we therefore derive, by substituting in equation (1) and in- 
troducing the value of R Q , 

x 1 x 3 x n n X n + J 

£*= 1+X+ + • + 



2 I-2- 3 1-2 ;/ 1-2 •■••(«+ I) 

Putting x equal to unity, we obtain the following series, which 
enables us to compute the value of the incommensurable quan- 
tity £ to any required degree of accuracy : 

e= I + 1 + — + 4- 

12 1-2-3 1-2-34 



I- 2-3 • • n 1-2-3 • • (n + 1) ' 

The computation may be arranged thus, each term being de- 
rived from the preceding term by division : 

2-5 
.16666666667 
4166666667 

833333333 

138888889 

I 984 1 270 

2480159 

275573 

27557 

2505 

209 

16 

I 



2.71828182846 



Since s 6 is less than £, the remainder (n being 14) is less than 
i\ of the last term employed in the computation, and therefore 
cannot affect the result. Inasmuch as each term may contain a 
positive or negative error of one-half a unit in the last decimal 



1 88 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 63. 

place, we cannot, in general, rely upon the accuracy of the last 
two places of decimals, in computations involving so large a 
number of terms. Accordingly, this computation only justifies 
us in writing 

B — 2.7l828l828. 



Logarithmic Series. 

164. The logarithmic series is deduced by applying Mac- 
laurin's Theorem to the function log(i + x). 
In this case 



/(■*■) = 


log(i + X) 


••• /(o) = 





A*) = 


I 

I + X 


••• /'(o) = 


I 


/"(*•) = 


I 


■•• /"(o) = 




(I + xf 




/"'(•*) = 


1-2 


••• /'"(o) = 


i-: 


/*(*) = 


1-2-3 


■• /"(o) = 


— 


log (I + 




^: 3 ^r 4 


+ 



i-2. 3, 

ence log (i + x) = x - - ^- + y ■ -y + • • ■ . . . (1) 

Since this series is divergent for values of x greater than 
unity (see Art. 158), we proceed, to deduce a formula for the 
difference of two logarithms, which may be employed in com- 
puting successive logarithms; that is, denoting the numbers 
corresponding to two logarithms by n and 11 + k, we derive a 
series for 

log (n -f k) — \ogn~ log — — . 



§ XXII.] 



LOGARITHMIC SERIES. 



I89 



A series which could be employed for this purpose might be 
obtained from (1), by putting in the form H — . We ob- 

'71 n 

tain, however, a much more rapidly converging series by the 
process given below. 

Substituting — x for x in (1), we have 



log (1 — x) = — x 



x 3 x i 
7 4 



(2) 



Subtracting (2) from (1), 



, 1 + x 

log = 2 

' 5 I - X 



XXX 

X + — + — + — + 

3 5 7 



(3) 



a series involving only the positive terms of series (1) 

' . 1 + x 11 + h , . h 

rutting = , we derive x = — 

B 1 — x 11 

in (3), we have 



, n + J 1 
5 11 



+ J 



h % 



+ * 



211 + h 



_2;^ + /^ (2;z + /^) 3 (2« + /^) 5 



; substituting 



•]• • (4) 



The Computation of Napierian Logarithms. 



165. The series given above enables us to compute Napierian 
logarithms. We proceed to illustrate by computing log e 10. 
The approximate numerical value of this logarithm could be 
obtained by putting n — 1 and h — 9 in (4) ; but, since the series 
thus obtained would converge very slowly, it is more convenient 
first to compute log 2 by means of the series obtained by put- 
ting n — 1 and h = I in (4) ; thus : 



log e 2=2 



+ 



I II II 

-4 ; + =4 



■333537 3 



■j 



190 



THE DEVELOPMENT OF FUNCTIONS. [Art. 165. 



We then put n — 8 and h = 2 in (4) ; whence 



l0g e IO=3 l0g e 2 + 



I 
+ - 

3 



+ 



■]• 



1 

-3 3*7 5 3 a 7 y 
In making the computation, it is convenient first to obtain 
the values of the powers of | which occur in the series for log 2, 
by successive division by 9, and afterwards to derive the values 
of the required terms of the series by dividing these auxiliary 
numbers by 1, 3, 5, 7, etc. The same auxiliary numbers are 



also used in the computation of log e 10. 
of the numerical work below. 



See the arrangement 



1 

3 


0.3333333333 


1 


0.3333333333 


iff 


370370370 


3 


123456790 


(if 


41152263 


5 


8230453 


(1)' 


4572474 


: 7 


653211 


(«" 


508053 


9 


56450 


(J)" 


56450 


11 


5132 


(IT 


6272 


13 


482 


(1)" 


697 


15 


46 


(§)" 


77 


17 


5 




log 


e 2 = 


0.3465735902 

2 




0.693 147 1 804 


1 
3 


0.3333333333 : 


I 


o.3333333333 


(I) 5 


41152263 : 


3 


13717421 


(i) 9 


508053 


5 


101611 


(i) 13 


6272 


7 


896 


a) 17 


77 


9 


9 



0.3347153270 
0.1115717757 

0.2231435513 
2.079441 5412 



3 l0g e 2 = _ 
l0g e IO = 2.30258509 



§ XXII.] THE COMPUTATION OF LOGARITHMS. 191 

166. The common or tabular logarithms, of which 10 is the 
base, are derived from the corresponding Napierian logarithms 
by means of the relation 

log e ,r = log e io \og 10 x, 
whence log IO ;tr = =— log e ,r = M. \og e x. 

The constant -. , denoted above by M. is called the modulus 

log. 10 ' y 

of common logarithms. Taking the reciprocal of log. 10, com- 
puted above, we have 

M — 0.43429448. 



The Developments of the Sine and the Cosine, 

167. Let f(x) = sin x, 

then 

f\x) = cos x, f"(x) = — sin x f /'"(x) = — cos x,f w (x) — sin x ; 

/ iv being identical with/, it follows that these functions recur 
in cycles of four ; their values when x = o are 

o, 1,0, — 1, etc. 

Hence substituting in equation (1), Art. 161, we have 

x 3 , x b x 1 , , 

sin x — x h • • • • • (1 ) 

1-2-3 1-2 • • • 5 1-2 • • • 7 V ' 

In a similar manner, we obtain 

x" x* x 6 

COS X = I 1 ! + . . . . . (2) 

1-2 1-23-4 1-2 • • • 6 ' 



I9 2 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 68. 

Huygens Approximation to the Length of a 
Circular Arc, 

168. A convenient formula known as " Huygens' Approxi- 
mation to the length of a circular arc " is derived by means 
of the series for sin^r.* 

Let s denote the length of the required arc, C its chord, c the 
chord of half the arc, and a the radius of the circle. Then 

C s . c s 

— = 2 sin — and - = 2 sin — . 
a 2a a 4a 

Developing by the series given in equation (1) of the preceding 
article, we obtain 



-£ = - ^_ + ^_ T _..,., ..(1) 

2a 2a 1-2.3-2 a* 1-2 • • • S' 2a 

. ess 2 s b ~ t \ 

and — = — — ■ + r^—i • • • . . (2) 

2a 4a i-2-3:2\a 1-2 •• • 5-2 a 

Multiplying equation (2) by 8, and subtracting equation (1) to 
eliminate the term containing s 2 , we have, approximately, 

8c — C _ 3«y 3 s 5 

2a 2a 4 1.2 • • • 5-2 5 # 5 ' 

, Zc-C s 5 , , 

whence — ^ = S ~j68c7^> (3) 

and, omitting the term containing s\ 

s = — ^— — 2c + i(2c — C). 



* This article is taken with slight modifications from Williamson's Differential 
Calculus, \Xi\t& edition, London, 1877. 



g XXII.] HUYGENS? APPROXIMATION. 193 

The error committed in adopting this value for s is less than 
the term containing s" in equation (3), since the remainder neg- 
lected in writing this equation is positive. 

Examples XXII. 

1. Expand (1 + x) m . 

(1 + x) m = 1 + mx -\ 2 l x * _| L ) v / ^3 + . . . 

1-2 I'2'3 

It is evident that no coefficient will vanish if m is negative or frac- 
tional. This is the form in which the binomial theorem is employed 
in computation, x being less than unity. 

2. Find three terms of the expansion of sin 2 x. 



. 


X 


2X' 


sin x = x - 


- — 


-j- 




3 


3'- 5 



3. Expand tan x to the term involving x b inclusive. 

X 3 2X° 

tan x = x H 1 f- • • ■ 

3 i5 

4. Expand sec x to the term involving x 6 inclusive. 

x~ sx* 6ix G 

sec x — 1 + 1 g H -f . • . 

1-2 I-2-3-4 1-2 • • • • 6 

5. Expand log sec x to the term involving x 6 inclusive. 

x- X* x° 

log sec x = 1 1 4- . . . 

2 12 45 

6. Find four terms of the expansion of £* sec jc. 

e* sec jc=i +^ + ^c"H (- • • • 

3 

7. Derive the expansion of log (1 — x 1 ) from the logarithmic series, 
and verify by adding the expansions of log (1 + x) and log (1 — x). 



194 THE DEVELOPMENT OF FUNCTIONS. [Ex. XXII 



8. Expand log(i — x + x 2 ) 

I + X ' 



I + x 3 
We may put i — x + x 2 in the form , and employ the loga- 



rithmic series. 



X 2X x x x 



log ( I — # + X 2 ) = — X -\ 1 1 

J 23453 

9. Derive the expansion of (1 + x)s* from that of £ x . 

(i + *)£*:= I + 2X + ^- 



1-2 



.% . 



10. Find, by means of the exponential series, the expansion of xe 2x , 
including the /zth term. 

£ x 

11. Expand by division, making use of the exponential 

series. 

€ x X 2 X 3 -IX* \\X* 

= 1+ + ^ +•'• 



1 + x 2 3 8 30 

12. Find the expansion of £ x log(i 4- x) to the term involving # 5 , 
by multiplying together a sufficient number of the terms of ' the series 
for e x and for log (1 + x). 



.2 a 



e x log (1 + x) — x + — + — + — + 
2 3 40 

13. Expand log (1 4- £ x ). 

2 4 

log(l + t -) = Iog2+-t. T -— +•• 

14. Expand (1 + s x ) n to the term involving x 3 inclusive. 

(i +€ x )"= 2 n \ I + ~ ■ X + -* 5— L ■ 

V ' ( 2 2 1-2 

;g(^ 2 + fl + 2) x 3 ) 

~r 3 +•••(" 

2 1-2-3 ) 



§ XXII.] EXAMPLES. 195 

15. Find the expansion of v / (i ± sin 2x), employing the formula 
|/( 1 ± sin 2.v) = cos x ± sin x. 

X* x % 
V( 1 ± sin 2x) = 1 ± x =F + • • • 

1-2 I- 2- 3 

16. Find the expansion of cos 2 x by means of the formula 
cos 2 x = i(i -H cos 2.*). 

2 3 * 4 2 5 * 6 

COS" 3: = I — X' -\ — - : + • • • 

1-2-3-4 1-2 • • • 6 

17. Find the expansion of cos 3 x, by means of the formula 
cos 3 x = J(cos $x + 3 cos jc). 

3* 2 3 4 + 3 * 4 , u 3* 1 + 3 * 5n 

cos 3 * = 1 — ^— +-— -^- • +(- 1) -. 

i-2 4 i'2-3-4 4 1-2 • • • 211 

18. Compute log e 3, and find log 10 3 by multiplying by the value of 
M (Art. 166). 

log e 3 = 1. 0986123. 
log IQ 3 = 0.4771213. 

19. Find log e 269. 

Put 11 = 270 = 10 x 3 3 , and h = — 1. 



20. Find log € 7, and log e i3. 



log e 269 = 5.5947 1 14. 

log e 7 = 1.9459101. 
log* 13 = 2.5649494. 



21. The chord of an arc, supposed to be circular, is found by meas- 
urement to be 5176.4 ft., and the chord of half the arc to be 2610.5 ft- > 
find the length of the arc by Huygens' approximation. 

s - 5 2 35-9 ft - 



I96 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 69. 

XXIII. 

Even Functions and Odd Functions, 

J69. Many simple functions have the property expressed by 
the equation 

/(-*)=/(*), ....... (1) 

and are called even functions : for example, cos;r, log(i -\-x 2 ), 
and a + x* + bx\ are even functions. 

Every rational integral function involving powers of x with 
even exponents only evidently has this property ; and it is also 
evident that the series corresponding to any function having 
this property can consist only of such terms. 

Certain other functions have the property expressed by the 
equation 

/(-*) = -/(*), (2) 

and are called odd functions : for example, sin x, tan x, and 
ax -f bx* are odd functions. It is evident that the series cor- 
responding to a function having the latter property can contain 
only odd powers of x. 

These two classes do not include all functions; in fact, it is 
evident that many functions satisfy neither (1) nor (2) : the de- 
velopments of such functions contain of course both odd and 
even powers. 

170. In the case of an odd function, which does not become 
infinite for x = o, we have, from (2), 

/(o)=-/(o) .-. 2/(0) = 0; 
therefore, when f(x) denotes an odd function, 
/(o) = o or /(o) = 00 . 



§ XXIII.] EVEN FUNCTIONS AXD ODD FUNCTIONS. 1 97 

The derivative of an even function is an odd function ; for, if 
we take the derivative of 

f{x)=f{-x), 

we obtain f\ x ) — ~~ f(— x \ 

In like manner, it may be proved that the derivative of an odd 
function is an even function. 

It follows that in the case of an even function all the deriva- 
tives of an odd order vanish or become infinite for x = o, and 
that in the case of an odd function the even derivatives vanish 
or become infinite. 



The Development of the Inverse Tangent. 

171. When the series of functions f f, f", etc., does not 
follow an obvious law of formation, the labor of deriving the 
terms of the development of f(x) by the ordinary process in- 
creases rapidly with the number of terms required. If how- 
ever the development of f'(x) is known that for f(x) may be 
found in the manner illustrated below. 

Let /(*) = tan -1 *, 

I 



then f\x) 



I +** 



which may be developed by division. Since the primary value 
of tan -1 ;r is an odd function, we assume for this value 

tan- 1 ^- = Ax + Bx' 6 + Cx* + Dx 1 +•--;... (1) 

by taking derivatives, we have 

— - a =A + sBx- + 56V + jDx" 4- • • • : 

1 ~\- St 



I98 THE DEVELOPMENT OF FUNCTIONS. [Art. 171 = 

but by division 

= I — X* + X* — X* + 

I + X 

These developments must be identical ; hence, we have 

A = i, B=-i, C = i, etc. 

and therefore, substituting in (1), 

tan -1 x = x — \x % + \x" — \x" + • • • 

When the series for f{x) is known, the series for f'{x) may 
be at once derived from it by differentiation. Thus the series 
for cos^r, Art. 167, may be obtained by differentiating the se- 
ries for sin x. 

The Development of Implicit Functions by Maclaurin s 

Theorem. 

172. The process of evaluating the derivatives in the case of 
implicit functions is exemplified below. A few terms only of 
the development are usually found in this way, as the process 
is necessarily tedious. 

Let f-6xy— 8=0; (1) 

whence, differentiating, 

(f-2x)^-2y = o, (2) 

•* <%y-4*+v-^=o (3) 

Putting x = o in equation (1), we obtain 

y — 2 (the only real value of y ), 



§ XXIII.] IMPLICIT FUNCTIONS. 1 99 



and, substituting this value for y in equation (2), we obtain 

cixAo 

and in like manner, by substituting in each of the successive 
differential equations the values of the derivatives already found, 
we obtain 



fin = fin ~ - 1 fin 



1, etc. 



Whence, substituting in equation (2), Art. 161, we have 



1 x x 

y — 2 + x - i • — + 



1-2-3 I-2-3-4 



Evaluation by Development in Series. 
173. When a function ^-7— ' takes the form - for x — o, its 

Q\X) O 

evaluation is frequently facilitated by developing the terms 
fix) and <f*(x) in series involving powers of x. 
Let the result of the development be 

f{x) _ A +A I x^-A 2 x- -l . . . 
4> (x) a Q + a 1 x + a 2 x' + -'-." 

Since, by hypothesis /(o) = o and 0(o) = 0,^3=0 and tf = o ; 
we may also find that one or more of the coefficients A z , A 2 , 
etc., and a x , a 21 etc. vanish. Let x m and x n be the lowest 
powers of x which remain in the numerator and in the denomi- 
nator respectively : then, 

/(*) _ *- _ A m +A„ M x + • • • 
<j>(x) x n a n + a n+1 x + • • • ' 



200 THE DEVELOPMENT OF FUNCTIONS. L Art - 1 73> 

When x = o, the second factor has the finite value — - ; hence 



<Kx)_ 



4±. ^1\ 

Cl n X U Jo' 



The value of this ratio is finite when its terms are of the same 
degree, but is either o or oo when these terms are not of the 
same degree. 

174. When this method is employed, we are sometimes able 
to determine at the outset the highest power of x which it is 
necessary to retain, as in the following example. 

Let it be required to determine the value of the function 

x sin (sin x) — sin 2 x~ 

x 6 _ 

It is unnecessary in this case to retain in the development of 
the numerator -any term whose degree is higher than the sixth ; 
and hence, in that of sin (sin x), no terms need be retained 
higher in degree than x\ Employing the series for sin x, Art. 
167, we have 

sin (sin x) = sin x — \ sin 3 x + T ^ sin 5 x — • • • 

= (x - \ x° + T y*-° ■ ■ •) - i (*• - \ x" . . .) + ^ x > ■ . . 

= X -3 X -f- yo X • • • ', 

and ^rsin (sin x) = x~ — \x^ + ^x e • • • ; 

also, squaring the series for sin x, 

sm*x = x* -\x* + &x e . • •; 
hence x sin (sin x) — sin 2 ;tr = -^x* • • • 

The value of the given fraction is therefore ^. 



§ XXII I.] /, VALl \1 770 . V B V DE J r EL OPM EN T. 20 1 



Had we found the development of the numerator to contain 
terms lower in degree than x\ the value of the fraction would 
have been infinite ; but had the term in x* vanished, the value 
of the fraction would have been zero. 

175. The method of development may also be used to fur- 
nish a demonstration of the ordinary formula for the evaluation 
of a function which takes the indeterminate form. 

By Taylor's theorem, using Lagrange's form of the re- 
mainder, we have 

f (a + h ) ^ f(a)+f'(a + dh)-h . 
4 (a + h) 4(a) + 4' (a + 6/i) ■ h ' 

when f(a) = o and <j>(a) — o, this equation reduces to 

f(a + //) f(a + Oh) . 
4(a + h) f(a + 6k) } 

hence, putting h = o, 

f(a)^f(a) 

4(a) 4 '(a)' 

a result identical with that obtained by a different process in 
Art. 95. 

Symbolic Form of Taylor s Theorem. 

7 11 

176. By employing the notation — , -— r , etc., for the de- 
rivatives, we may write Taylor's theorem in the form 

/(,+/,) =/(,)+* £/(*)+£ • £j(^~r 3 ■ £/w+ ■ • • 

The form 

jrr , i\ id /l* d* h % d* , "1 /-/ \ 



202 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 76. 

in which f(x) is removed from within the brackets as if it 
were a factor, may also be employed, inasmuch as this form 
of writing the second member admits of no ambiguity of mean- 
ing. It will be noticed that the terms within the brackets pro- 
ceed according to the law of the exponential series (Art. 163) ; 
we may therefore write 

f(x + h) = ***/(*), 

h- 
which is to be understood as implying that £ * d * is to be devel- 
oped in accordance with the form of the exponential series, 
and f(x) introduced after each term. 



Examples XXIII. 

1. Prove that each of the following expressions is an even function 
of x ; viz., — 1 

sin x 2 , (sin -1 .*) 2 , £~^. 

2. Prove that each of the following expressions is an odd function 
of x ; viz., — 

I + X , € x + I 

log — , and — . 

& 1 — x ' s x — 1 

3. Prove that the following functions are even ; viz., — • 

x , x 
x cot x, and 



2 £' — I 

4. Prove that the following functions are odd ; viz., — 

log tan (Jtt + x), and log [V{i + x Q ) + x]. 

5. Prove that, when (/>(x) is an even function, f[<l>(x)] is likewise 
an even function, / denoting any function. 

6. Prove that, if f(x) and (f>(x) are both odd functions, f[<f>(x)~\ 
is an odd function. 



§ XXIII.] EXAMPLES. 203 

7. Deduce the expansion of log (1 + x) by the method employed 
in deducing the expansion of tan -1 x in Art. 171. 

8. Deduce the expansion of e x by means of an assumed series. 
Compare the series for f'(x) with the series assumed for f(x). 

9. Expand sin -1 x by the method of Art. 171. 

The expansion of f\x) may be effected by the binomial theorem. 

sin-\* = *-f i-i* 3 + i f-K + M-l ■*** + * ' • 

10. Derive by differentiation three terms of the expansion of tan* 
from that of log sec x [Ex. XXII, 5], and from the result derive the 
expansion of sec 2 x. 

11. Expand e ex to the term containing x* inclusive. 

We easily deduce f'(x) = £ x f(x) ; therefore, the product of the as- 
signed series and the exponential series must be equivalent to the derivative 
of the former. 

£< x = £ [1 + X + X* + \x* + • • •] 

12. Given s v + xy = e, to find the expansion of y in powers of x. 

xix 2 1 x % 

J € 12 6 1-2-3 £ 

13. Given y 2 — xy = i, to find the expansion of y in powers of x. 

1 x" 2 2 x* 

y=±i +£x± — - — =F 4 ± • • • 

^ 2122 1-2-3-4 

14. Given _y = 1 + x e\ to find the expansion of y in powers of x. 

x* x* 

y = I +£x+2£ 2 + ge 3 - + . . . 

12 y 12-3 

Evaluate the following functions by the method of Art. 173. 

m sin — sin mQ , . « 

\c. —. — — - , when Q .= o. §m. 

Q (cos — cos mh ) 



204 THE DEVELOPMENT OF FUNCTIONS. [Ex. XXIII. 

, i (i + x) log(i + x) 

10. — 2 

(x + sin 2.# — 6 sin -l-^;) 2 
J 7 



18. 



(4 + cos x — 5 cos £#) 8 ' 
tan 7nxr — nx 



2x tan 7ro: 



6 (20 + sin 20 —4 sine) 

10. •, 

3 + cos 20—4 cos 



when ^ = 0. 


-i 


x = 0. 


8.2 9 2 

3* ' 


a- = 0. 


7T 2 

6 " 


= 0. 


-f 



XXIV. 

Z!^£ Development of Functions by Means of 
Differential Equations. 

177. When a general expression for the /zth derivative of a 
function cannot be obtained, it is, nevertheless, frequently pos- 
sible to derive the particular values which the derivatives take 
when x = o, by the method explained in Articles 91 and 92. 
Thus, in the case of the function 

f(x) = y — sin (m sin- 1 .*-), 

by substituting in Maclaurin's series the numerical values of 
the derivatives when x = o, as derived in Art. 92, we have 
[since f(o) = o] 

sin (m sin- 1 ^) = #ar 4- — ^ t x * H * J -^- L x • ■ • 

This expansion is equivalent to that of sinms in powers of 
sin ,2 ; for, if we put z = sin- 1 .*, the equation becomes 



sin mz 



. r ;# 2 — 1 . 2 , (;/* 2 — 1)(/^ 2 — 9) . 4 "I / T \ 

=msmz\ 1 sin 2 ,^ ^ — ^.^sin 4 ^ . (1) 

L 1-2-3 1-2.3.4.5 J 



§ XXIV.] BY MEANS OF DIFFERENTIAL EQUATIONS. 205 

This series will consist of a finite number of terms when m is 
an odd integer. 

In a similar manner, it may be proved that 

11 f . a , m*( th* — 4) . . , N 

cos viz = 1 — — sin" z -\ - — — — sin 4 #—•••,. . (2) 

1-2 I-2-3-4 V J 

the number of terms being finite when m is an even integer. 

178. The complete process in the case of the function 
(sin- 1 ^) 2 is given below, sin -1 x here denoting the primary 
value of this function. 

Let y — (sin- 1 ^-) 2 , . (1) 

, dy 2 sin- 1 x , x 

then -7- = -77 n> ( 2 ) 

dx V(i — x-) v } 

x sin- 1 x 

, d 2 y V{i — x') , 



Combining (2) and (3), to eliminate the transcendental functions 
and radicals, we obtain the differential equation 

, 2 . d~y dy 

Taking, by means of Leibnitz' theorem, the ^th derivative of 
each term (see Art. 88), we have 

/ ~d"**y d n+x y , .d n y d n+1 y d n y 

( 1 —x 2 ) -, — ~ — 2n x -= — f- — n(n—i) -r^ — x -, — ■?- — n -~- — o, 
' dx n+z dx n+I } dx n dx^ 1 dx' 1 

whence putting x = o, we derive 

dx»^ dx- 1 v J 



206 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 78. 

Since from equation (2) we have 

dy 



■£\r °' 



equation (4) shows that the odd derivatives all vanish for x = o, 
and also enables us to determine the values of the even deriva- 
tives, the value of the second derivative being first obtained 
from equation (3), thus : — 



d% y _■„ * k y " „. d *y 



1 .2 



dx> *> = 2 - 2 ' dS= 2 " 2 - 4 ' etC - 

Finally, substituting in equation (2), Art. 161, we obtain the 
expansion, 



(sin- x ;r) 2 = 2 



x^ 2V 2 2 -4 2 .r 6 2 2 -4 2 > 6 2 x B _ \ n 

1-2 I-2-3-4 ' 1-2 • • • 6 1-2 • • • 8 J 



;tr 2 2 ^ 4 2-4 x 6 2-4-6 ;tr 8 



•} 



The Co7nputation of tt. 

179. By differentiating the series for (sin- 1 ;?) 2 , obtained in 
the preceding article, we derive a convenient series for the 
computation of n ; viz., — 

sin- 1 ;? 2 3 ■ 2-4 B . 2-4-6 7 

= ;r + - ^ H - ;T H ^— X + • • • 



V(i -^ 2 ) 3 3-5 . 3-57 

This equation gives an expression for the circular measure of 



§ XXIV.] 



THE COMPCTAT/OX OF n. 



207 



an angle in the form of a series involving powers of its sine ; 
for, putting sin- 1 ,!' = 6, we have 



6 — cos 6 sin 6 



+ ? sin 2 + 2 -^ sm*8 + • . 
3 3-5 



The arc may also be expressed in terms of its tangent ; 
thus, since sin 6 = tan 6 cos 0, 



6 = 



tan 



2 tan'fl 2^4tan 4 <9 



sec 2 L 3 sec 2 3-5 sec 4 # 
or, denoting tan 6 by x, 



■] 



tan- 1 ^ = 



.r 



I + JT"" L 



2 X 2-4 

I + r + 



3 1 + x" 3.5 \l -h* 



• (0 



(2) 



180. This series is convergent for all values of x. Since 
tan \n = I, we can obtain a series for \it by making x = I. 
We are however enabled to use much more convergent series 
by employing the formula, 

- = 5 tan- 1 - + 2 tan- 1 A . . . . 
4 7 79 

* The employment of this formula in the computation of Tt was suggested by 
Euler in a memoir on the subject in 1779. 

This, as well as many similar formulas which have been used for the same pur- 
pose, may be readily verified by means of the expression for the tangent of the sum 
of two arcs. Denoting the tangents of these arcs by m and », the tangent of the 
sum is 

m + n 



whence, putting m = -f, and n = -■■%-, we obtain 

tan- 1 I + tan- 1 - a 9 - = tan" 1 & . 
Hence 5 tan -1 1 + 2 tan -1 -f 9 - = 3 tan -1 f •+- 2 tan -1 -fc , 

in like manner the second member may be reduced to 

tan -1 ^ + 2tan _1 £ , and finally to tan -1 1 or \iC. 



208 



THE DEVELOPMENT OF FUNCTIONS. [Art. l8o. 



For x — \ y the fraction 



and, for x = -fa, 



I + X* 50 100 



X 



144 



1 + x 2 6250 1 00000 



The small values of these fractions render the series for 
taj|- r i and for tan- 1 T 3 ¥ rapidly convergent, and the computa- 
tion is facilitated by the fact that the denominators are powers 
of ten. 

181. Substituting the equivalent series for the inverse tan- 
gents in equation (2) and multiplying by 4, we have 



n — 



10 



22 24 
3 100 35 



-)'■ 

00/ 



+ 



30336 
1 00000 

2 4 



1 + 



44 Y 



3 5 \ 100000/ 



144 



3 1 00000 



This series may be written in the following form, in which 
each of the letters a, /3, y, etc., denotes the value of the term 
preceding that in which it occurs. 



71 = 



14 
15 



4 16 / 2 Y 6 2a 8 2/? , 11 

— + — - — + + + - 

00 10 \ioo/ 7 100 9 100 1 



+ 



10112 
1 00000 



+ *** + 4 . J44«_ + 5 . I44P 
iooooo' 5 1 00000 7 1 00000 



2y 
I OO 



■] 



••} 



The numerical work for ten places of decimals is given 
below. The method by which each term is derived from the 
preceding term is indicated by the latter form of writing the 
series : multiplication by the fractions f, f , etc., is effected by 
deducting 1 J, etc., of the quantity to be multiplied. 



§ XXIV.] THE COMPUTATION OF n. 209 



a — 


3 -04 
0.0006.4 


[ 

28 

182857 


0.00 


a = 
144 a = 


3- 

: 0.00288 

41472 
82944 


ft = 


] 


097H3 

21943 

2438 


0.00144/? = 


331776 
478 

68 


r = 




19505 
390 




r = 


: 410 




3.00288332186 


s = 




35 

355 
7 
1 






30028833219 

300288332 

30028833 

60O5767 


€ = 




6 




7t ~ 


0.30365 1 561 5 1 




3.040651 17009 
.20271007801 

2.83794109208 


2.83794109208 

: 3.I4I5926536 



Examples XXIV. 

1. By means of the series given in Art. 177, derive values of sin 5^ 
and of sin yx. 

sin $x = 5 sin x — 20 sin 3 x -f 16 sin x ; 

sin yx = 7 sin x — 56 sin 3 x + 1 1 2 sin 5 x — 64 sin 7 #. 

2. By means of the series given in Art. 177, derive values of cos 4X 
and of cos 6x. 

cos 4X = 1 — 8 sin 2 jc + 8 sin 4 x ; 

cos 6x = 1 — 18 sin 2 .%• + 48 sin 4 jc — 32 sin 6 x. 

3. By means of formula (2), Art. 177, prove that 

cos z = 1 — \ sin 2 z — \ sin 4 z — T ^ sin 6 z — • • • 



210 THE DEVELOPMENT OF FUNCTIONS. [Ex. XXIV. 

4. By means of formula (2), Art. 177, derive four terms of the ex- 
pansion for cos \z. 

cos \z = 1 — \ sin 2 z — T j-g- sin 4 z — T | \± sin 6 z — • • • 



5. Expand tan * - by means of the 72th derivative of this function. 

(See Art. 87, and compare Ex. XII, 20.) 

3 5 

1 X X X X 

tan - = 5 H 1 — - ' - 

a a 2> a 5 a 

6. Expand £ z cos x, by means of the nth. derivative. (See Art. 85.) 

3 2 4 2 5 

r 2* 2 * 2 X 

€ x COS a: = I -f # 

1-2-3 i-2 • • • 4 1-2 • • • 5 

3 7 4 8 

2 X 2 X 



1-2 • • • 7 1-2 • • • 8 



7. Expand £ s sin#. (See Ex. XII, 6.) 

£ sin # = # + # + 



1-2-3 1-2 • • • 5 1-2 • • • 6 1-2 -.. 7 

8. Expand ^ cosa cos (* sin <*). (See Ex. XII, 8.) 

^ 2 

e xcosa cos (# sin «) = 1 + cos a • # + cos 2 a • • • 

v ' 1-2 

9. Write the expansion of sin~\#, employing the results found in 
Ex. XII, 19. 

. . 1 x 5 1-3 x b i m V1 x 

sin" 1 * = x + + _^._ + _A_| + . . . 

23 2-45 2-4-6 7 

10. Write the expansion of log [x 4- V(a* + «* 2 )], employing the 
results found in Ex. XII, 21. 

io g [, + YK + ,')]=io g » +s *-^^;^-... 



^ XXIV.] /..WIMPLES. 211 

ii. Write the expansion of [x + V{a' + x n ')] m , employing the re- 
sults found in Ex. XII, 22. 

a 
\x + Via* + -r 2 )]'" = a w + ma'-'x + — a^V 

L 1-2 

1-2-3 I-2-3-4 

12. Expand by Taylor's theorem the function 

a cos [log (1 + //)] + b sin [log (1 + //)], 
employing the results obtained in Ex. XII, 24. 

a cos [log (1 -f /&)] -f £sin [log(i + h)~\ =a + bk— - tf 

-\a + b 7 „ /£ 4 
+ ^ ^ — 100 + • • • 

1-2-3 1-2 • • • 4 

13. Expand f msin " 1 - r by the method of Art. 178. 

«f« n -i* w 9 .* 8 ;;z(;//* + 1) - m*(m 2 + 4) * , 

g« sin -^ — j _j_ mx _] _| V _ / ^3 + _V ^W_ x * + , . , 

1-2 1-2-3 I-2-3-4 

14. Given the differential equation 



rt 7 y dy 



and j> = a, 

to expand y in powers of x. 



L 4 4'9 4'9* 16 4-9 ,l6 ' 2 5 J 



212 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 82. 

XXV. 

Functions of Imaginary Quantities. 

182. Functions of imaginary variables are in general imagi- 
nary quantities capable of expression in the form 

a + bi, 

in which i — V{— i). 

The transformation of a function of an imaginary variable 
to the above form may often be effected by developing it into 
a series, and separating the real and imaginary parts. 

Thus, to transform the function £ ix , we put ix for x in the 
exponential series; since i= V{— i) gives 

z' 2 = — i, i 2 = — i, z 4 = I, etc., 

the result may be written in the form 

x 2 ix % x* 
e* = i + tx + + • • • , 

1-2 1-2-3 I-2-3-4 



or e tx = 



_.£l + _f* T 

1-2 1-2-3-4 J 

••]■ 



3'4 

x s 



t-2-3 I-2-3-4-5 

The series in brackets are the developments of cos;r and of 
sin^r respectively [see Art. 167]; we have therefore, by sub- 
stitution, 

&*= cos^ 4- i sin;r (1) 

If the independent variable be of the form a + bi, the de- 
velopment of the function will contain powers of a + bi, and 



£ XXV.] FUNCTIONS OF IMAGINARY QUANTITIES. 213 

may therefore, by expanding these powers, be put in the form 
a 4- bi. 

183. Every imaginary quantity may be expressed in the 
form p(cos6 -f- /'sin 6) ; for, if we put 

a 4- bi — p(cos 6 4- isinff), (1) 

the equations, a — p cos and b — p sin 0, . . . (2) 

give^ p = i/(V 4- b'), and tan = — . . . . (3) 

Equations (3) always give real values for p and 6. The positive 
quantity p is called the modulus, and the angle 6 is called the 
argument. The latter admits of but one value between o and 
2t, the quadrant in which it terminates being determined by 
the signs of a and b [see equations (2)] ; denoting this value by 
0\ all the values of 6 are included in 

6=2k7Z + &, 

k denoting zero or any integer. 

184. The imaginary quantity a + bi can be expressed in the 
form pe' 9 ; for by equation (1), Art. 182, we can write 

a 4- bi — p(cos 6 -f i sin 6) — pe' 9 . . . . (1) 

in which p and S are determined as in the preceding ar- 
ticle. 

From equation (1) we have 

log (a 4- bi) = log p 4- id = log p 4- i{6' 4- 2kn\ . . (2) 

hence every imaginary quantity has an unlimited number of 
logarithms ; the value of the logarithm obtained by putting 
k = o is regarded as the primary value. 



214 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 84. 

Putting b = o we have, when a is positive (see Art. 183), 

p = a and & = o .*. log a — log a + 2k ni; 

whence it follows that log a may be regarded as a multiple- 
valued function having but one real value ; viz., the primary- 
value as defined above. 

To obtain the logarithm of a real negative quantity we put 
b = o and — a for a ; whence p = a and 6' = n. 

log (— a) = log a + (2k + i)tt/, 

the primary value being log a + ni. 

185. By changing the sign of i in equation (1) we obtain 

a — bi~ p £~ z ' 9 , 

in which the modulus is unchanged while the sign of the argu- 
ment is reversed, a + bi and a — bi are called conjugate imagi- 
nary expressions. Their sum is the real quantity 2a, and their 
product is the real positive quantity a' + &. 

Hyperbolic Functions. 

186. Exponential expressions for sin^r and cos^ may be de- 
rived from equation (1), Art. 182 ; thus, putting — x for x in 

e ix _ CQS x _|_ I g j n x ^ 

we have e~ ix = cos x — i sin x ; 

whence cos x = J(V-* + e- ix )> (1) 

and sin^r = — r{e {x — e- ix ) (2) 



§ XXV.] HYPERBOLIC FUNCTIONS. 21 5 

Again, putting ix for x in these expressions for sin^r and cos^r, 
we have 

cos ix = \ (e* + *-*), (3) 

sin £r = U(a x — e~ x ) (4) 

The value of cos ix is real and is called the hyperbolic cosine of 

x, and the real factor of the expression for sin ix is called the 

hyperbolic sine of x. These real quantities are generally denoted 

s 1 n h x 
by cosh .r and sinh x, and the ratio — - — by tanh^r. Thus, 

cosh x 

we have cosh x = \{e x + e~ x ) 



sinh x = 


J(e* 


— f 


tanh;r = 


€- v - 


- £~- r 


e x ■+ 


■ f-' r 



The reciprocals of these quantities are denoted by sech x, 
cosech x, and coth x, respectively. 

Formulas involving these functions and presenting a re- 
markable analogy to the trigonometric formulas may readily 
be deduced. (See Ex. XXV, 7, 8, 9, 10.) 



De Moivres Theorem. 

187. Putting mx for x in equation (1), Art. 182, we have 

gmx — cos mx -j- i sin mx ; 

and, since e imx = (e zx ) m , 

cos mx + z sin ///.r = (cos x + i sin .r) w ... . (i) 

This result is known as -De Moivres Theorem^ and is understood 
to mean that, when the second member of the equation is ex- 



2l6 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 87. 

panded, the value of the real part will be cos mx, and that of 
the real factor of i will be sin mx. 

(88. Expanding the second member of equation (1) by the 
binomial theorem, we obtain the expansions for sin mx and 
cos mx in terms of sin x and cos x ; the number of terms in each 
of these expansions will be finite when m is a positive integer. 

Thus, putting m — 3 in equation (1), we obtain 

cos 3-3T + Z sin 3^=cos 3 ^ + 3cos 2 ^r- i sin x—3 cos^;- sin 2 ;?— zsin 3 ;tr. 
Whence cos $x = cos 3 x — 3 cos x sin 2 x> 

and sin ^x = 3 cos 2 ;tr sin x — sin 3 x. 

Multiple Values of the nth Roots of Real and 
Imaginary Quantities. 

I89. Expressions for the roots of quantities in the form 

p(cos 6 + z'sin 6) 

may be derived by means of De Moivre's theorem. Putting 

- for m in equation (1), Art. 187, and employing the general 

value of 6 given in Art. 183, we obtain 

[p(cos6> + ^ sin 0)]* = p"( cos ■ h*sin J. . (1) 

When n is an integer, by giving k the values o, 1, 2, . . . {n — 1), 

we get n values of the arc ■ ■ less than 27r; every other 

n 

value of this arc will differ from one of these by a multiple 

of 27T. The second member of equation (1) will therefore have 

n and only n different values. 



§ XXV.] APPLICATIONS OF DE MOIVRE'S THEOREM. 217 

190. Making p = 1 and 6' — o, we have for the values of the 
;/th root of unity the n values of the expression 

2k 7T . . 2k 71 t N 

cos h i sin (2) 

n n 

The real root, unity, is obtained by putting k — o. The imagi- 
nary root corresponding to k — 2 is, by De Moivre's theorem, 
the square of the one that corresponds to k = I. If the latter 
is denoted by a, the 71 roots may be expressed thus, — 

I, a, of, a . . . . a' 1 - 1 . 
The roots a and a"- 1 are conjugate roots ; for 

2(ll — \)7t . . 2{n — iW 27T . . 27Z 

a"- 1 — cos -^ — 4- 1 sin -^ — = cos 1 sin — , 

71 71 71 n 

whence (compare Art. 185) 

27T . 

^ + a"- 1 = 2cos- — , and aa n - 1 =i. 

71 

In like manner all the other roots occur in conjugate pairs, 
with the exception of the real root 1 corresponding to k — o 
when 7i is odd, and of the real roots 1 and — 1 corresponding 
to k — o and k = \n when 71 is even. 



The Resolution of Certain Expressions into Factors 
by means of De Moivres Theorem. 

191. The expressions derived in the preceding article being 
roots of the equation 

x n — \ — o, 
we have 

, v / 271 . . 271 

x n — I = (x — 1 ) [x — cos i sin — ■ 

y \ n n 

2(71— iW . . 2(71— \)n\ 
x — cos — zsin— f — J. . . . (1) 



2l8 THE DEVELOPMENT OF FUNCTIONS. [Art. 191. 



The product of the factors corresponding to each pair of 
conjugate roots is a real quadratic factor. Thus, 

(x — a) (x — a n ~ T ) = x 2 — (a + a"- 1 ) + a n — x* — 2x cos hi. 

Combining in this way the conjugate factors we have, when n 
is odd, 

/ \ / -i 27T 

x n — I = \X — I) I X — 2X cos + I 



and when n is even 



(x* — 2x cos n + 1 ) , . . . (2) 



^ w — I = (x — i) (x 2 — 2X COS f- I J 



x' 1 —2XCOS 7t + I \(X + l). ... (3) 



192. To resolve x 2n — x n cos 6 + 1 /«^ factors, n being an 
integer, we solve the equation 

x™ — x n cosO + 1 = o (1) 

From (1) we obtain 

x n — cos 6 ±i sin ^. 

Hence, expressing the values of x by means of equation (1), 
Art. 189, we have 

2k n + & , . . 2k n + (9' ,\ 

^ — C os ± 2 sin , . . . (2) 

the values of k being o, 1, 2, . . . (n — 1). The product of the 
factors corresponding to the two conjugate roots determined 



§ XXV.] APPLICATIONS OF DE MOIVRE'S THEOREM. 219 

by a single value of /- in equation (2) reduces to the real quad- 
ratic expression 

2k 7T + 6' 
X' — 2X COS \- I. 

n 
Hence we have 

& \/ , 27T+6' 
X 2 " — 2X N Q0SU-\- I = ( X — 2^T COS hi X —2X COS ■ + I 

n J \ n 

2(n— i)/t + 0' \ 
;tr — 2,r cos — * - f- 1 J . . . (3) 



The Sine and the Cosine Expressed as Continued 
Products. 

193. Putting x = I in equation (3), the first member be- 
comes 2(1 — cos 6), and each factor in the second member 
takes a similar form. The positive square root of 2(1 — cos 6) 
is 2 sin \& , since \& is between o and n. Taking in like man- 
ner the square root of each factor in the second member, we 
obtain 

. ' . 6' . 27t + 6' . 47t + 0' . 2 (n-\)7i + 6' t . 

2 sin Id —2 n sin — sin sin- • • sin— . (1) 

211 2n 2n 2n 

The above equation is however true for all values of ; for, 
if we add 27T to 6\ the first member changes sign; but in the 
second member the first factor assumes the present form of the 
second factor, the second that of the third, and so on ; finally 
the last factor becomes equal to the present value of the first 
factor with its sign changed ; therefore the second member also 
changes sign. Hence & may have any value, and putting 6 in 
place of \& we have 

sin 6 = 2—sin^ sin (* + 6 -) sin (™ + 
n \n nJ \ n 

. An — i)7t 
sm f- '— + -) (2) 



220 



THE DEVELOPMENT OF FUNCTIONS. [Art. 1 93. 



The last factor may be written in the form 

. (7t d\ 

sin ( ), 



n 6 

sin hr 1 — 

n n 



hence for the product of the second and last factors we have 



71 



7t 6\ . (7t 

sin ( — I — sin ) = sin" sin' 

nj \n nj n n 



In a similar manner the third factor and the last but one may- 
be combined ; therefore, if n is an odd number, we have 



• n . 6 { . „7Z . . 

sin u = 2 n ~ I sin - sin sin 

n \ n 



sim 



27t 



sin — 
11 



/ . 3 « — 1 

( sin 



7t . ^ 

sin — 

n n 



■ (3) 



Dividing this equation by sin 6 and then making 6 = o, we have 



2 n ~ x . q 7T . ^in 

I — sin — sin — 

n n n 



. a (n — i)7t 

sin 2 -^ '— , » 

2n 



and dividing equation (3) by the last equation, we have 



sin 6 = nsin 



sin 



1 — 



L 



sin' 



1 — 



• J' 
sin — 
n 



snr 



27T 

n J 



sin - 
n 



. 9 (n—i)7e 

sin v — 

2n 



Finally in the above equation if 6 remains fixed, and n increases 
without limit we have, on evaluation, 



sin d = 6 



1 — 



e 2 



2V 



y "1 



(4) 



the number of factors being unlimited. 



§ XXV.] FACTORS OF THE SINE AXD THE COSINE. 22 



194. A similar expression for cos 6 may be derived from 
equation (4) of the preceding article, by means of the formula 



whence 



cos 6 



26 



cos 6 



sin 20 
2sin# 



"-£)(-5)(-5)(-i£ 



J? 

471' 



and removing common factors, we have 



cose/ = 



1 — 



unr. 



4^1 

9 n- 



25 7l'. 



• • • (5) 



By expanding the continued products in equations (4) and 
(5) and equating the coefficients of given powers of 6 to the 
corresponding coefficients in the expansion given in Art. 167, 
certain numerical series involving powers of n may be derived. 
Thus, equating the coefficients of tir in the expansions of sin 6 y 
we derive 



I I 

I + — + — + 

2- 3 



71 

6"" 



Bernoulli s Numbers. 

195. A series of numbers, first employed by James Bernoulli 
in 1687, present themselves in the expansions of certain func- 
tions, among which is the function 



\x 



+ I 



222 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 95. 

This expression is easily shown to be an even function ; 
hence if we make the transformation 



X 

and write y = — , (2) 



the development of y will contain no terms involving odd 
powers of x except the term — \x. 

196. To determine the values of y and of its derivatives 
when x — o, we have from equation (2) 

e*y-y = X., . . . . ;. . . . (3) 

and differentiating s* -f + e x y f. = 1 (4) 

ax ax J 

By applying Leibnitz' theorem (Art. 88) to equation (3), we de- 
rive, for values of n greater than unity, 

d H y r d n ~ x y n(n-i) ^d n ~ 2 y d n y .. 

dx n dx n 1 2 dx 71 - 12 J dx n KJ/ 

putting x = o in (4), we have y Q — 1 ; therefore equation (5) be- 
comes, when x = o, 



d n -y~] ^ n(n - 1) d n ~% 

dx n -*Ao 1-2 dx n ~ 2 J Q dx_ 



y 



dy' 



+ 1 - o. . (6) 



From equation (6) the values of the derivatives may be found 
by putting n — 2, 3, 4, etc. Thus making n — 2 we have 



dy 
dx 



1-1- 



§ XXV.] BERNOULLI'S NUMBERS. 223 

but, since the development of y contains no odd powers of x 
except the first, the remaining odd derivatives vanish. The 
numerical values of the even derivatives are Bernoulli's Num- 
bers ; the notation adopted being 

m d n y 



*« = -<-')' *. 



(7) 



because, as will be shown in Art. 199, the values of 

B t , B 2 , B 3 , etc., 

thus defined, are all positive. The development of y may 
therefore be written 



I - 


-* + A* -A x 

2 1.2 1.2. 3-4 

x 6 




°I-2 • • • 6 4 I-2 



£* — I 

(8) 



Putting 271 + 1 in place of n in equation (6), adopting the 
notation of equation (7), and introducing the values of the de- 
rivatives already determined, we obtain 

-(- 0"(M + 1)2?. + (- !)- (*»+ *»(2« - I) J?,,, . . . 

1-2-3 

(2« + 1)271 „ 2/z — 1 
+ i '- — B 1 = o , 

1-2 2 

and, solving for B n , 

A=a »*L=JL *„ - n (*» - W" - *)(*" ~ 3) B „ + . . . 
3 3-4-5 

+ (- i)^A - (- 1)" : 



2(272 + I) 

Substituting for ;/ the successive values 1, 2, 3, etc., we derive 

-£>i = f, 25 2 = -y-g-, B 3 = -fa y ±> A = -30 , ZJ 5 = -g^g- , 



224 THE DEVELOPMENT OF FUNCTIONS. [Art. I97. 



The Development of cot 0. 

£ x + I 

197. The development of the even function \x — ; differs 

from that of y in equation (8) of the preceding article only in 
the absence of the term containing the first power of x (see 
Art. 195). This function remains real when ix is substituted 
for x ; for we have 

Ux-r-— — = i — B T — — B 2 — B 3 -p — • • • 

■ e zx — 1 1-2 I-2-3-4 .1-2 • • • 6 

But (see equations (1) and (2), Art. 186), 

A . &* '+ I , . & ix + £-**"■* < • 2 cos kr 1 . . 

4-fcT -: = \lX — ^- = 4^ =-=— * — = 4* cot J* ; 

2 £ « _ j 2 &x _ £ -*« 22 Sill \x i * 

hence putting for \x 

? 2 ft 1 ? 4 i 2 6 <9 6 

1-2 I-2-3-4 1-2 • • • 6 

198. A series for cot in which the coefficients of the 
powers of assume a different form is obtained from equation 
(4), Art. 193, by taking logarithmic derivatives ; thus, 

20 20_ 20_ 

' a I 7? 2 a > 3" J if 

cot6 = -e w~ w w. ; 



whence #cot 



20" f 2 \~* 20 2 



The expansion of each term after the first in the second mem- 
ber is of the form 

92 I __ e i_Y 1 - # 2 ( 2 0" \ 

rrV ' k*7t*J '~ ~ 2 kW V + k*** + k*n> + ' ' ')' 



k 
in which k has the successive values 1, 2, 3, etc. 



XXV.] THE DEVELOPMEXT OF 6 COt 



225 



Substituting these expansions and collecting the coefficients of 
the powers of tf, we have 



6 cot 6 = 1 - 2 



.- + * + 4,-.] 



I I #" 

+ _ + + ...j_ 2 _ 



i + 2 -,+ 



'I" 



199. Comparing this expansion with (1) of Art. 197, we 
obtain 



A 



1-2 7T 2 



I I 

+ — + — 

2 3 



^ 2 



and in general B n 



Hence we have 



2.3.4 71* 
i-2---2n Ti 



+ ^ + Y 



I I 

I _| (_ — 

2 2W X 291 



etc., 



I+ _L + J_..., = (?«)-*«_, (I) 

2 2 « 3 2W 2 • 1-2 • • • 2;/ 7 

which serves to determine the sum of an arithmetical series of 
the above form in terms of the Bernoullian numbers. For in- 
stance when n = I, we have the result already obtained in Art. 
194, and when n — 2, we have 



I + A + 4- + 
2 4 3 



7t 
90 



Equation (1) may also be written in the form 
2 • 1 «2 • • • 2n 



B n = 



{27ty 



1 2 2W 7. 271 



which shows that Bernoulli's numbers are all positive; and also 



226 THE DEVELOPMENT OF FUNCTIONS. [Art. 1 99. 

that they increase rapidly with n ; for, as n increases B„ ap- 
proaches but always exceeds 

2 • 1-2 • • • 2H 



[27ty n 

Examples XXV. 

tan _ Vx = J/ log 



1. Prove that 

1+1 



1 — .%■ 

»&£ ^r/. 171 and equation (3), ^4r/. 164. 77^ real factor in this ex- 
pression for ta.n~ x ix is the inverse hyperbolic tangent of x. 

2. Prove that 

sin~7x = /log [x 4- Y(i + x 2 )], 

by means of the results obtained in Ex. XXIV, 9 and 10. 

The real factor in this expressio?i for sin~7.# is the inverse hyperbolic 
sine of x. 

3. Prove that 

[ |/(i — x~) + /*]'" = cos [z« sin -1 .*] + z sin [;/z sin"V] = e misin ~ x . 
See Ex. XXIV, n, and Art. 177. 

4. Find the primary value of log/ (see Art. 184.) 

log i = \in. 

5. Prove by means of the result obtained in the preceding example 
that 

6. Derive the fundamental formulas of Trigonometry by transform- 
ing the identity e ix e iy = e™**'^ by means of equation (1), Art. 182. 

7. Prove the relations : — 

cosh 2 A- — sinh 2 ^ = 1, and sech 2 ^ = 1 — tanh 2 .*. 

8. Prove the relations : — 

sinh 2* = 2 sinh x cosh x, and cosh 2X = cosh 2 x + sinh 2 x. 



£ XXV.] EXAMPLES. 227 

9. Prove the formulas : — 

sinh [x ± y) = sinh x cosher ± cosh x sinhj, 
and cosh (x ±y) = cosh * cosh y ± sinh* sinh j». 

10. Prove the formulas : — 

*/(sinh x) — cosh xdx, //(cosh x) = sinh x dx, //(tanh x) = (sech*)V*, 

d(sech x) = — sech x tanh*//*, //(cosech*) — — cosech x coth x dx, 

//(coth *) = — (cosech x)~ dx. 

11. Find expressions for sin 5* and cos 5* by means of De Moivre's 
theorem. 

sin 5* = 5 cos 4 x sin x — 10 cos 2 x sin 3 x 4- sin 5 *, 
cos 5* = cos 5 * — 10 cos 3 x sin 2 x 4- 5 cos * sin 4 x. 

12. Find expressions for sin;;/* and cos;;/* by means of De Moi- 
vre's theorem. 

m(m — 1) _ 

cos ;;/* = cos'"* b cos'" ~x sin" x + ■ • • 

1-2 

;;/(;// — i)(;;/ — 2) 
1-2-3 

13. Find the sixth roots of unity. 

The roots are ± 1, ± (h ± - 4/3). 

2 

14. Find the fourth roots of — 1. 

The roots are ±1-4/2(1 ± i . 

15. Find the cube roots of i. 

The roots are — /, ± J 4/3 4- hi. 

16. Resolve * n 4- 1 into factors. 
When n is even, 

* n + 1 = ( * 2 — 2* COS - + I J ( * 2 — 2* COS — + I ) • • • 

( 2 n — l \ 

( * — 2X COS 7t 4- I I ; 

when ;z is odd, 

* n + I = 1 X~ — 2* COS h I ) • • • ( X 1 — 2* COS /T — 1 ) ' * 4- 1) 

\ n / \ n J 



228 THE DEVELOPMENT OF FUNCTIONS. [Ex. XXV. 

17. Express the hyperbolic sine in the form of an infinite product 
by putting ix for Q in equation (4) Art. 193. 

sinh* = x (r + £)(! + ^-) (x + ^ • • ■ 

18. Express the hyperbolic cosine in the form of an infinite product 
by putting ix for Q in the expression for cos x derived in Art. 194. 

cosh "=( I+ f ! )( I+ 3^)( I + ^)"- 

19. By means of equation (8), Art. 196, and the relation 



find the development of 



s x + 1 
x 



e x + 1 



-(jH.^^ y-i) ,, , K-i) , + „, 



=■-.-£* 



£ x + I 2 1-2 I-2-3-4 1-2 • • -6 

20. From the 
development of 



20. From the result obtained in the preceding example, derive the 

a x — 1 



e* + 1 



£? — I 2 2 



l)^ x 2(2'-!)^^ , [ 2(2 6 -l)^ 3 / 



£* + I 1-2 I-2-3-4 1-2 • • • 6 

or, employing the values of B x , B 2 , etc., given in Art. 196, 

s x — 1 _ x x* _x^_ 17 x' six 9 

£ x + I ~~ 2 24 240 4032O 725760 

21. By means of the identity 

tan x — cot # — 2 cot 2#, 
derive the development of tan# from equation (1), Art. 197. 

tanx = Bi A^A x + J?2 ^^A^ +B3 ?^iA^ + 

1-2 I-2-3-4 1-2 • • • 6 



§ xxv.] 



EXAMPLES. 



229 



22. By means of the identity 



cosec x = cot cot x, 

2 

derive the development of cosec x from equation (1), Art. 197. 

1 r, 2 ( 2 — 1) ^ 2 ( 2 * — 1) a 
cosec x —- + B T — x + B* — ' x % + • • • 

X 1-2 I'2'3«4 

23. By putting = ttjc and taking logarithmic derivatives derive 
from equation (4), Art. 193, the series, 



7T COt 7T X = - ~\ 1 1 1 ; \- ■ 

X X — I X + I X — 2 X-\-2 X — 3 



+ ' 



24. By a method similar to that employed in the preceding example, 
derive the series 



n 1 

— tan n x = 

2 1 — 



2X I + 2X 



2X $ + 2X 



+ 



25. By taking derivatives of the series obtained in example 23 and 
putting x = J, derive the numerical series : — 



7T* 


III 

— +-;- + — + 
3 5 _ 7 2 


7T 3 
3 2 


111 

7 + 7"7 r + 


7T 4 

96 


7 + W + 



CHAPTER VIII. 

Curve Tracing. 



XXVI. 

Eqttations in the Form y = f (x). 

200. WHEN a curve given by its equation is to be traced, 
it is necessary to determine its general form especially at such 
points as present any peculiarity, and also the nature of those 
branches of the curve, if there be any, which are unlimited in 
extent. 

The general mode of procedure, when the equation can be 
put in either of the forms, y =f(x) or x — <j) (y), is indicated in 
the following examples. 

Asymptotes Parallel to the Coordinate Axes. 

201. Example I. c?y — x 2 y = a 3 (i) 

Solving for y, we obtain 

y = -* — -s (2) 



When x = O f y = a. Numerically equal positive and nega- 
tive values of x give the same values for y; the curve is there- 
fore symmetrical with reference to the axis of y. As x increases 



§ XXVI.] ASYMPTOTES TARALLEL TO THE AXES. 



23 



from zero, y increases until the denominator, a' — x\ becomes 
zero, when y becomes infinite ; this occurs when x — ± a . 

Draw the straight lines x — ± a. These are lines to which 
the curve approaches indefinitely, for we may assign values to 
x as near as we please to + a or to — a, thus determining points 
of the curve as near as we please to the straight lines x — a and 
x — — a. Such lines are called asymptotes to the curve. 

When x passes the value a,y becomes 
negative and decreases numerically, ap- 
proaching the value zero as x increases 
indefinitely. Hence there is a branch 
of the curve below the axis of x to 
which the lines x — a and y = o are • 

asymptotes. 



The general form of the curve is in- 



Fig. 16. 



dicated in Fig. 16. 

The point (o, a) evidently corresponds to a minimum ordi 
nate. 



202. Example 2. a*x=y(x — a)* '. . (1) 

Solving for y, 



y 



a~x 



i* - «r 



(2) 



When x is zero, y is zero ; y increases as x increases unti] 
x = a, when y becomes infinite. Hence 



is the equation of an asymptote. When x passes the value 
a, y does not change sign, but remains positive, and as x in- 
creases y diminishes, approaching zero as x increases indefi- 
nitely. 



232 CURVE TRACING. [Art. 202. 

Examining now the values of y which correspond to nega- 
tive values of x, we perceive that, y becoming 
negative, the branch which passes through the 
origin continues below the axis of x, and that 
y approaches zero as the negative value of x 
Fig. 17. increases indefinitely. Hence the general form 

of the curve is that indicated in Fig. 17. 

. 203. To determine the direction of the curve at any point, 
we have 

, dy . a 4- x f N 

tan <f> = -f- = — a" - ^ (3) 

Y dx (x - ay vo; 

The direction in which the curve passes through the origin 
is given by the value of tan <j> which corresponds to x = o. 
From (3), we have 

dy~ 
ax J 

the inclination of the curve at the origin is therefore 45 . 



Minimum Ordinates and Points of Inflexion. 

204. To find the minimum ordinate which evidently exists 
on the left of the axis of j, we put the expression for -j- equal 
to zero, and deduce 

x = — a. 

The minimum ordinate is therefore found at the point whose 
abscissa is — a ; its value, obtained from equation (2), is 

-la. 

d*y 
A point of inflexion is a point at which -—^ changes sign (see 



§ XXVI.] POINTS OF IX FLEX ION. 233 

Art. 76) ; in other words, it is a point at which tan $ has a 
maximum or a minimum value. In this case there is evidently 
a point of inflexion on the left of the minimum ordinate. From 
equation (3) we derive 

d y _ , x + 2a 

d?~ 2a (x-aY 

putting this expression equal to zero to determine the abscissa, 
and deducing the corresponding ordinate from (2), we obtain, 
for the coordinates of the point of inflexion, 

x = — 2a, and y = — \a. ■ 



Oblique Asymptotes. 

205. Example 3. x % — 2xy — 2x- —Sy = o (1) 

Solving this equation for y, we have 

x~ x — 2 . N 

y= ^- x ^- A (2 > 

It is obvious from the form of equation (2) that the curve 
meets the axis of x at the two points (o, o) and (2, o). Since 
y is positive only when x > 2, the curve lies below the axis of 
x on the left of the origin, and also between the origin and the 
point (2, o), but on the right of this point the curve lies above 
the axis of x. 



206. Developing the second member of equation (2) into 
an expression involving a fraction whose numerator is lower in 
degree than its denominator, we have 

y = hx-l+2 2 ~J C - (3) 

X + 4 



234 



CURVE TRACING. 



[Art. 206. 



The fraction in this expression decreases without limit as x in- 
creases indefinitely ; hence the ordinate of the curve may, by 
increasing x t be made to differ as little as we please from that 
of the straight line 

y=ix-i. 



This line is, therefore, an asymptote. 




The fraction 



2 — x 



x> + 4 
is positive for all values 
of x less than 2, negative 
for all values of x greater 
than 2, and does not be- 
come infinite. The curve, 
therefore, lies above the 
asymptote on the left of 

the point (2, o), and below it on the right of this point, as 

represented in Fig. 18. 



Fig. 1 



207. There is evidently a minimum ordinate between the 
origin and the point (2, o). 

We obtain from equation (2) 



dy 
dx 



x 2 + I2.r — 16 

(* 2 + 4) 2 ' 



(4) 



and 



d'y _ -x*+6x 2 + \2x — 8 
dx 1 ~ (x 2 + 4) 3_ 



dy _ 



• (5) 



Putting •—- = o, we obtain x = o and x — 1.19 nearly, the only 

real roots; the abscissa corresponding to the minimum ordinate 
is therefore 1.19, the value of the ordinate being about — 0.11. 
The root zero corresponds to a maximum ordinate at the 
origin. 



XXVI.] 



OBL IQ I '/■ .IS) \ MP TO TE S. 



235 



Puttin 



d\v 



- o, wc obtain the three roots x — — 2, and 

x = 2 (2 ± v 7 3) ; the corresponding ordinates are obtained 
from equation (3). There are, therefore, three points of in- 
flexion, one situated at the point (—2, — 1), and the others 
near the points (0.54, — 0.05), and (7.46, 2.55). 

The inclination of the curve is determined by means of 
equation (4) to be tan _I J at the point (2, o), and tan _I f at the 
left-hand point of inflexion. 



Curvilinear Asymptotes. 

208. Example 4. x 3 — xy + 1 == o (1) 

Solving for y 



y 



= X* + 



(2) 



In this case, on developing^ in powers of x, the integral portion 
of its value is found to contain the second power of x ; the 
fraction approaches zero when x increases indefinitely ; hence 
the ordinate of this curve may be made to differ as little as we 
please from that of the curve 

y = s (3) 



The parabola represented by this equation 
is accordingly said to be a curvilinear asymp- 
tote. It is indicated by the dotted line in 
Fig. 19. 

209. The sign of the fraction - is always 

the same as that of x, and its value is infinite 
when x is zero ; hence the curve lies below 
the parabola on the left of the axis of y, and 
above it on the right, this axis being an 
asymptote, as indicated in Fig. 19. 




\ 

Fig. 19. 



236 CURVE TRACING. [Art. 209. 

Taking derivatives, we obtain 

i= 2x ~h • •••••• (4) 

g-Hi) (*> 

There is a point of inflexion at ( — 1, o) ; the inclination of the 
curve to the axis of x at this point is tan _I (— 3). 

There is a minimum ordinate at the point (i^/4, i^/2). 

This cubic curve is an example of the species called by- 
Newton the trident. The characteristic property of a trident 
is the possession of a parabolic asymptote and a rectilinear 
asymptote parallel to the axis of the parabola. 



Examples XXVI. 

1. Trace the curve y = x (x 2 — 1). 

Since y is an odd function of x, the curve is symmetrical with re- 
ference to the origin as a centre. Find the point of inflexion, and 
the minimum ordinate. 

2. Trace the curve y 2 (x — 1) = x 2 . 

The curve has for an asymptote the line x— 1 ; there is a mini- 
mum ordinate at (2, 2), and a point of inflexion at (4, IV3). 

3. Trace the curve f — x 2 (x — a), determining its direction at 
the points at which it meets the axis of x. 

The asymptote is found by the method of development, thus 

1 Y—x{\ ^— etc. J; 

x) \ sx 9X 1 

the equation of the asymptote is therefore 

y — x — \a. 



§ XXVI.] EXAMPLES. 237 

4. Trace the curve x" + xy 4- 2x — y = o. 

5. Trace the curve y~ = x' + x 3 , and find its direction at the 
origin. 

The curve has a maximum ordinate at (— f, ±f 1/3). The 
value of y 2 may be taken as the function whose maximum is required. 
(See Art. 130.) 

6. Trace the curve 7 = x 3 — xy. Find the point of inflexion, the 
minimum ordinate, and the asymptotes. 

The curve has a rectilinear asymptote x = — 1, and a curvilinear 
asymptote^ =jc 2 — x + 1. This curve is a trident. (See Art. 209.) 

7. Trace the curve y 2 = x 3 — x*. 

Both branches of the curve are tangent to the axis of x at the 
origin. 

8. Trace the curve y* —4y = x 3 + x*. 
Solving for y, we obtain 

y •= 2 ± V{X 3 + X* + 4) = 2 ± V[{X + 2) (X* - X + 2)]. 

The factor x 2 — x + 2 being always positive, the curve is real on 
the right of the line x = — 2. 

Find the points at which the curve cuts the axis, and show that 
the upper branch has a maximum ordinate for x = — -| and a mini- 
mum ordinate for x — o. 

9. Trace the curve (x — 2d) xy = a(x — a) (x — 3^). 

10. Trace the curve (x — 2d) xy' = d (x — a) (x — 3^). 

11. Trace the curve y* = x* (1 — x*) 3 : find all the points at which 
the tangent is parallel to the axis of x. 

12. Trace the curve 6x (1 — x) y = 1 + 3^. 

This curve has a point of inflexion, determined by a cubic having 
only one real root, which is between — 1 and — 2. Find the three 
asymptotes, and the maximum and the minimum ordinate. 



238 CURVE TRACING. [Ex. XXVI. 

13. Trace the curve x*y* — (x + .2)* (1 + x 2 ). 
Solving the equation forjF, we have 



y 



±^^+^)=±(^+^)(i+y. 



The asymptotes are y = x -f 2,y = — x —2, and x =0. The 
curve has a minimum ordinate corresponding to a • = ^2 ; the inclina- 
tion at the point at which it cuts the axis of x is tan -1 ( ± \ 1/5). There 
is a point of inflexion corresponding to the abscissa x = — 6.1 nearly. 

14. Trace the curve (y 2 — £ 2 ) 2 = a s x ; determine the points at 
which tan (f) is infinite ; and, denoting by x Q the abscissa of the point 
at which the curve cuts the axis of x, show that the abscissa of the 
two points of inflexion is f x > 

15. Trace the curve y 2 (x — a) = x 3 + ax 2 . 
Solving for j', we have 

or y — ± (x + a) 1 1 -\ ^ + etc. J . 

The asymptotes are x = a, y = x -\- a, and y = — x — a. There is a 
point of inflexion corresponding to x = — 2a. 

16. Trace the curve y 2 x* = d~ (x~ + y 2 ). 
The asymptotes are x —- ± a, and y — ± a. 

17. Trace the curve x* ( v — x) + xy + x" + x + 2 = 0. 

Show that the equation of the oblique asymptote is y = x — 2. 
Find the points in which the curve cuts the axis of x. 

18. Trace the curve (x — k) y — (x — a) (x — b) (x — c). 
Taking a> b> c, trace the curve, first for k < c ; secondly, for 

k > c and < b. Examine also the intermediate case in which k = c. 

19. Trace the curve (x — k) y 1 = (x — a) (x — b) (x — c). 



>> XXVI.] EX. \MPLES. 239 

Taking a > b > c, trace the curve, first for k < c ; secondly, for 
k> c and < b. Examine also the cases in which /' = c,a = b, and 
a = b — c. 

20. .Trace the curve (y — x 1 ) 2 — x = o. 

Both branches of the curve touch the axis of x at the origin. The 
lower branch cuts the axis of x, has a point of inflexion, and a maxi- 
mum ordinate. 

21. Trace the curve jv 3 — axy — b'x = o. 

This curve is a trident (see Art. 209), having a point of inflexion 
at the origin. 



XXVII. 

Equations in the Form f (x, y) = o. 

210. In this section and the next we illustrate by examples 
some of the methods which may be employed in tracing alge- 
braic curves, when it is not possible to solve the equation so as 
to make one coordinate an explicit function of the other. 

Example 5. x % — y 3 — x 1 + 2j/ 2 = o (1) 

The points in which this curve cuts the axes are readily 
obtained ; thus, putting 

x = 0, 

we have y 1 {2 — y) = o ; . (2) 

the roots of this equation are o, o, and 2 ; hence the origin 
and the point B (o, 2) are points of the curve (see Fig. 20). 
Putting 

y = c, 

we have x' 1 {x — 1) — o; (3) 



240 CURVE TRACING. [Art. 210. 

the roots are o, o, and I ; hence the point A (i, o) is a point of 
the curve. 



The Employment of an Auxiliary Variable. 

2(1. The intersections of the curve with the line 

y = mx 

are determined by the equation obtained by substituting this 
value of y in equation (i) ; viz., 

x z (i — m 3 ) —x* (i — 2iff) = o (4) 

Since x 1 appears as a factor of the first member, x — o is a 
double root of this equation, whatever be the value of "m ; hence 
every line of the form y — mx is said to meet the curve in two 
coincident points at the origin. In fact, there are in this case 
two branches of the curve passing through the origin, and this 
furnishes an explanation of the double roots found in equa- 
tions (2) and (3). 

212. The curve may now be traced by means of the follow- 
ing expression for the third root of equation (4), which is the 
abscissa of P, the only point distinct from the origin in which 
the line y — mx cuts the curve. 

Rejecting the factor x\we obtain, from equation (4), 

1 — 2m" 1 , . 

* = - rx> • (5) 

1 — m 

and, since y — mx, y =. ■ — r (o) 

1 — m 

Regarding ^ as a variable, (5) and (6) express the coordi- 
nates of any point of the curve. 



XXVI I.j EMPLOYMENT OF AN AUXILIARY VARIABLE. 241 




Fig. 20. 



Putting in = o, the line y = mx coincides with the axis of x, 
and (5) gives x — 1, showing that P 
coincides with A. 

As ;// increases from zero x de- 
creases, and becomes zero when m = 
4/A, the point P describing the upper 
part of the loop AO. The position 
of the line y = mx, corresponding to 
m — V£, is indicated by the dotted 
line in Fig. 20. This line is a tangent 
to the curve at the origin. 

As m increases from this value to 
unity, x becomes negative and passes to infinity, when OP 
makes an angle of 45 with the axis of x\ P describing the in- 
finite branch in the third quadrant. 

As m passes the value unity, x again changes sign, and 
decreases ; the point P describing the infinite branch in the first 
quadrant, passing through the point (1,2) when in — 2, and 
arriving at the point B (o, 2) when m = 00. 

As the line OP passes into the second quadrant, m becomes 
negative and decreases numerically ; x also becomes negative, 
and returns to zero when m = — V^ ; P describing the branch 
BO. 

Finally, as m passes through this value and returns to zero, 
x becomes positive and increases to unity ; P describing the 
branch OA in the fourth quadrant, and returning to the point 
whence it started. 

If it be desired to construct the curve with accuracy, as 
many points as we choose may be determined by assigning 
values to m, and computing x and y by equations (5) and (6). 

The method of determining the asymptote drawn in Fig. 20 
is explained in Art. 228. 



242 CURVE TRACING. [Art. 2 1 3. 

The Direction of the Curve, 
213. From equation (1) we derive 
dy __ $x' 2 — 2x 



dx 3/ - 4j/ 



= tan j (7) 



This expression, involving both x and y, may be employed to 
determine the direction of the curve at any point whose coor- 
dinates have been determined ; for example, the inclination of 
the curve to the axis of x at the point (1,2) is tan -1 \. 

The derivative -5- , which takes the form - at the origin, 
dx O 

gives, when evaluated by the method of Art. 117, two values 

of tan (j> which are identical with the two values of m for which 

the point P passed through the origin. It is to be remarked 

* dv 
that the expression for the derivative -— must necessarily take 

the indeterminate form at a point through which two or more 
branches pass, otherwise tan <f> would have but a single value. 

214. The numerator in the expression [equation (7)] for 
tan (J) vanishes for x — o and for x = § ; hence tan (f> = o for 
these values of x, except when the denominator vanishes at 
the same time. Thus, corresponding to x — o, we have a maxi- 
mum ordinate at B, while at the origin both terms of the frac- 
tion vanish. Since the cubic equation which results from the 
substitution of x — f in equation (1) has three real roots, neither 
of which reduces the denominator of the expression for tan $ 
to zero, there are three points on the line x = § at which tan (j) 
vanishes. 

In like manner, the denominator vanishes when y = o and 
when y — f . Accordingly, corresponding to y — o there is a 
maximum abscissa at the point (1,0). Putting y — % in equa- 
tion (1), we obtain a cubic equation which has two imaginary 



g XXVII.] THE DIRECTIOX OF THE CURVE, 243 

roots, and one real root equal to —0.81 (nearly) ; there is 
therefore a minimum abscissa corresponding to y = $, whose 

value is — 0.81 approximately. 

215. Example 6. x* — ^axy- + 2ay s = o (1) 

Putting j' = mx as in Art. 211, we have 

x i — (5am 2 — 2am \ .r 3 = o (2) 

This equation has three zero roots ; hence, whatever be the 
value of ///, the line y = mx is said to cut the curve in three 
coincident points at the origin. 

Rejecting the factor x 3 from equation (2), we 
have 

x — iam- - 2am 3 .... (3) 




Fig. 21. 



for the abscissa of P, the only point distinct from 
the origin in which the line y — mx cuts the curve ; 
and for the corresponding ordinate we have 

y = ytm* - 2am 4 (4) 

When m = o, equation (3) gives x = 0, P being at the 
origin. 

As ;;/ increases from zero, x becomes positive, but returns to 
zero when m — f, while P describes the loop in the first quadrant. 

As tn increases from § to oc, x passes from o to — oo, while 
P describes the infinite branch in the third quadrant. 

As 111 changes sign and passes from — cc too, x becomes 
positive and y negative, and their numerical values decrease 
to zero, while P describes the infinite branch in the fourth 
quadrant and returns to the origin. 

Maxima Values of the Coordinates Expressed in Terms 
of an Auxiliary Variable. 

216. In finding maxima and minima ordinates and abscissas, 
it is often convenient to treat y and x as functions of m. Thus, 
from (3), we obtain 



244 CURVE TRACING. [Art. 2 1 6. 

-— = 6am — 6am" 1 ; 
dm 

putting this derivative equal to zero, we derive 

m = o, and m = i. 

//z = i gives the point (#, #) at which there is a maximum ab- 
scissa. 

In like manner, from (4), we derive a 

--j— — gam 2 — Sam 3 ; 
dm 

putting this derivative equal to zero, we obtain 

m = o, and m — f. 

By substituting the value m — J in equations (3) and (4), we 
obtain, for the coordinates of a point of the curve which has a 
maximum ordinate, 

x — o.g$a and y — 1.070. 

The abscissa corresponding to m =0 is a minimum ; but 
the ordinate is neither a maximum nor a minimum, as may be 
seen by examining the motion of P as the revolving line OP 
passes through the position OX, Fig. 21. 

Tangents and Points of Inflexion. 

217. If we suppose a straight line to pass through a given 
point of a curve, and to rotate about this point, then, as the 
line approaches the position of the tangent through the given 
point, it will have a second intersection with the curve, and 
this second intersection will coincide with the given point 
when the rotating line reaches the position of the tangent. 
For this reason a tangent line is said to meet the curve in two 
coincident points at the point of contact. If however the given 



§ XXVII.] TANGENTS AND POINTS OF INFLEXION. 245 

point on the curve is a point of inflexion, the rotating line will 
evidently have two other intersections, one on each side of this 
point, and the three intersections will finally come into coin- 
cidence. Hence a tangent at a point of inflexion is said to 
meet the curve in three coincident points. 

218. When the point of contact is at the origin, the value of 
tan <f) at that point, is proved in Art. 1 1 8 to be the same as that of 

— in the equation resulting from placing equal to zero the 

terms of lowest degree in the equation of the curve. Thus, in 
the case of the curve traced in Art. 202, the equation may be 
written in the form 

x*y — 2axy -+- # 2 (y — x) = O ; . . . . . ( 1 ) 
the above method gives 

y-x-o\ (2) 

whence at the origin tan <j>= 1. 

But since equation (2) may be regarded as the equation of a 
line in the form y —■ mx, in which m denotes the value of tan ({> 
at the origin, (2) is itself the equation of the tangent at the origin. 
If now we combine equations (1) and (2) so as to eliminate y, 
the terms of the first degree will vanish, and the result will 
contain x 1 as a factor ; whence we have the double root x = o, 
indicating the two coincident points at which the tangent 
meets the curve. 

219. If the given equation contains no terms of the second 
degree, or if these terms are divisible by those of the first degree ', 
the tangent at the origin will meet the curve in more than two 
coincident points. Thus, if the equation of the curve be 

x % + xy* + 4r 2 — y -f x — y — o, 



246 CURVE TRACING. [Art. 2 1 9. 

the tangent at the origin, 

y — x t 

will cut the curve in three coincident points, since the equation 
obtained by eliminating y from the given equation will con- 
tain the triple root x* — o. Hence this curve has a point of in- 
flexion at the origin. 

220. In like manner if, while the equation of the curve con- 
tains terms of the first degree, but no absolute term, the lowest 
terms which do not vanish with the terms of the first degree are 
of a degree higher than the third, the tangent at the origin is 
said to meet the curve in more than three coincident points. 
Since however there is in this case but a single branch of the 
curve at the origin, it is impossible for more than two real 
points to come into coincidence with the given point, hence we 
infer that some of the intersections of the rotating line with the 
curve are imaginary ; and, since imaginary roots occur in pairs, 
we have a point of inflexion when the number of coincident 
points is odd but not when that number is even. 

Nodes and Cttsfts at the Origin. 

221. When the equation of a curve contains no terms lower 
in degree than the second, it may be shown, as in Art. 211, 
that every straight line passing through the origin meets the 
curve in two coincident points. The curve is in this case said to 
have a node at the origin. 

Since a tangent at an ordinary point of a curve meets it in 
two coincident points, the tangent to either branch of a curve 
at a node meets the curve in three coincident points. 

222. The values of tan ^ at a node are found by the process 
employed in Art. 218. In the case of the curve 

J 3 -/-/ + 2/=0 (I) 



§ XXVII.] NODES A. YD CUSPS AT THE ORIGIN. 2tf 

constructed in Fig. 20, Art. 212, the equation thus obtained is 

X*-2y'=0 (2) 

The two values of - deducible from this equation are identical 

with those of tan <j> at the origin ; hence, regarding x and y as 
variables, this equation is satisfied by the coordinates of any 
point on either of the tangents at the origin ; in other words, 
equation (2) is equivalent to the equations of both tangents at 
the origin. Resolving equation (2) into factors, we have 

x -\- y ^/2 — o and x— y\/2 = o, 

the separate equations of the two tangents at the origin. 

The above reasoning is obviously general ; hence we infer that 
whenever the curve passes through the origin, the equations of 
the tangents at that point may be obtained by placing equal to 
zero the terms of lowest degree in the equation of the curve. 

' • v 

223. When, as in the above example, the values of - are 

x 

found to be real, we infer that the curve has two branches 

passing through the origin which in this case is called a cru- 

node. 

When on the other hand the values of — are imaginary, the 

x 

curve can have no real branches passing through the origin, 
and the curve is said to have a conjugate point or acnode. 

Finally, if the two tangents are found to be real and coin- 
cident, there will usually be two real branches of the curve 
having a common tangent at the origin and terminating at that 
point, which is in this case called a cusp. When the branches 
lie on opposite sides of the tangent the cusp is called -zceratoid* 
cusp, and when they lie on the same side it is called a ram- 
phoid-f cusp. 

* From xepa<3, a horn. f From pdjiiqioS, a beak. 



248 CURVE TRACING. [Art. 224. 

224. It is to be remarked that in some cases when the given 
equation seems to indicate a cusp at the origin, this point will 
nevertheless be found to be an isolated point, y being imagi- 
nary for values of x near zero. For example, the equation 

x G - *V -ay = o 

indicates the existence of two tangents at the origin coincident 
with the axis of x ; but, on solving for y, we obtain 

whence it is obvious that y is imaginary for all values of x nu- 
merically less than a, except zero, the origin is therefore an 
isolated point. 



Multiple Points. 

225. When the equation of a curve contains no term lower 
in degree than the third, the curve is said to have a triple point 
at the origin. In general, when an equation contains no terms 
lower in degree than the mth., the curve is said to have a mul- 
tiple point of the mtk order at the origin, and the equation 
which results from putting equal to zero the terms of lowest 
degree is the equation of the m tangents at the origin. Thus, 
resuming example 6, Art. 215, the equation of the curve is 

x* — $axy* + 2ay z = o, 

whence the equation of the tangents at the origin is 

3X/-2/ = 0; 

therefore the tangents are 

y — o, y — o, and $x — 2y = o. 




§ XXVII. ] MULTIPLE POINTS. 249 

The cusp in Fig. 21 corresponds to the double root y — o, the 
axis of x being the common tangent to the two branches ; the 
other tangent is indicated by the dotted line. 

226. A triple point is regarded as equivalent to tJiree nodes ; 
for suppose three branches of the curve to pass very near 
to the same point, as in Fig. 22, these branches 
will intersect in three nodes. If, now, one of these 
branches be so moved as to cause the three nodes to 

coincide, these points may be regarded as uniting to 

r i. • 1 • i. Fig. 22. 

form a triple point. 

In like manner, a multiple .point of the order m is equiva- 
lent to \m (111 — 1) nodes ; this being, by the algebraic theory 
of combinations, the number of intersections that ultimately 
coincide. 

The triple point in Fig. 21 is equivalent to two nodes and 
a cusp. 

The number of real branches which exist at the origin may, 
however, be less than ;/z, since one or more pairs of the tan- 
gents may be imaginary (see Art. 223) ; and, also, because one 
or more pairs of the branches corresponding to equal values 
of m may be imaginary (see Art. 224). Thus a point which 
fulfils the condition requisite for a triple point may have but 
a single branch of the curve passing through it, and may con- 
sequently present to the eye nothing to distinguish it from an 
ordinary point of the curve. 

Examples XXVII. 

1. Trace the curve y s — x 3 — y 4- 4X = o. 

Since the equation contains terms of odd degrees only, if a point 
(x, y) satisfies the equation, the point (— x, — y) will likewise satisfy 
it ; hence the curve is symmetrical in opposite quadrants. It follows 
that the asymptote is symmetrically situated, and must therefore pass 
through the origin. The intersections of the curve with the axes and 
with the lines y = ± 1, and x = ± 2 are easily obtained; also the 



250 CURVE TRACING. [Ex. XXVII. 

lines on which the points having maxima and minima ordinates are 
situated. 

2. Trace the curve y 4 — i6x 4 + x 1 — 4xy — o. 

Find the points at which the curve cuts the axis of x and its direc- 
tion at these points. 

3. Trace the curve 2j 5 — ^xy* + x 5 = o. 

Find the maximum ordinate and the maximum abscissa by means 
of the derivative expressed in terms of x and y. The former occurs 
at the point (^2, ^4), and the latter at (^27, ^3). 

4. Trace the curve x h — $xy — $xy 2 + y b = o. 

Find the asymptote, and the tangents at the origin. Determine 
points on the loops by making m = 1, and find the direction of the 
curve at these points. 

5 . Trace the curve x l — axy 1 + y i = o. 

The tangents to the loop in the first quadrant are parallel to the 
axes at the points ( J«, \d) and (J ¥3 .a, J ^27 . «). 

6. Trace the curve jc 4 — ax*y + ^y 3 = o. 

Find the points at which the curve is parallel to the coordinate 
axes. 

7. Trace the curve x 4 + x*y* + y 4 — ax (x* — y 2 ). 

The minimum abscissa is x = (1 — -f V3) # ; the maxima and mini- 
ma ordinates are at the points determined by x = (i ±. yV 4/2 1) # and 
y = ± \ V3 • a- 

8. Trace the curve x 4 — y 4 — cfxy = o. 
y — x and jy = — x are asymptotes. 

9. Trace the curve a: 4 — 2a*x* — 2 ay 3 4- 30^* = o. 
The value of x in terms of m may be put in the form 

x = am 2 ± a (m* — 1) V(m' + 2). 

The curve has nodes at the points determined by m = ± 1. Deter- 
mine the direction of the curve at these points, and determine also the 



sj XXVI I.] EXAMPLES. 251 

points at which the curve is parallel to the axes, by means of the de- 
rivative in terms of x and y. 

10. Trace the curve y* + x* —Say' + 6axy = o. 

The curve is symmetrical with reference to the axis of y and 
consists of three loops. The minimum ordinates correspond to 
m = ± \ 1/3. 

11. Trace the curve (-v 2 + y')' = 4.T 2 + r. 

The origin is a conjugate point. To find the maximum ordinates 
employ the expression for v 2 in terms of m. vi = 00 gives the points 
(o, ± 1) ; and ;;/ = ± V2 gives the points (± J V6, ±f ^3). 

\2. Trace the curve (x~ + y~) 3 — 4tfVy 2 ~- o. 

The curve consists of four loops. A maximum ordinate occurs at 
(fa t 6, %a 1/3). 

13. Trace the curve y* — g6ay~ + iooa' 2 x- — jc 4 = o. 

Find the coordinates of the points at which the abscissa has maxi- 
ma and minima values. In constructing the curve, determine the 
points corresponding to m = J and m — 2. This curve is known as 
" la c our be du diab/e." 

14. Trace the curve a 9 (x + y)~ = {a 2 — x"') b . 

The curve has two cusps at both of which the line x + y = o is 
tangent to the curve. Find the direction of the- curve at the points at 
which it cuts the axis of y. 



XXVIII. 

Points at Infinity. 

121 , When the generating point of a curve recedes to in- 
finity, the value assumed by the ratio - may be determined by 
the following method. 



252 CURVE TRACING. [Art. 227. 

Let the equation of the curve be assumed in the general 
form 

Ax 11 + Bx n ~ x y • • • + A'x n ~ x + • • • + A"x n ~ 2 • • • = o, . . . (1) 

in which the first group of terms is of the nth. or highest de- 
gree, the second of the (n— i)th degree, etc. Dividing the 
equation by x n we have 

x x L • x _ 



Now when x and jy are simultaneously infinite, assuming that 

- has a finite value, this equation reduces to 

x 

A + £* + . . . = o, (2) 

The equation 

Ax n + Bx n ~ x y+ • •• = o, (3) 

of which the first member consists of the group of terms of 

v 
highest degree in equation (1), determines values of - identical 

x 

with those determined by equation (2). If the value of- is 

x 
infinite when x and y are infinite, the reciprocal ratio - will 

have a finite value ; therefore by dividing by y n it may be shown 
that in this case also equation (3) gives the required values of 
the ratio of x and y. 

In discussing the general equation, we shall frequently use 
the abridged form 

Pn + Pn-i + ' ' ' .+ Po = O, 



XXVIII.] 



POIXTS AT INFIX I TV. 



253 



in which P„ indicates the sum of the terms of the nt\\ degree, 
P„_ x , the sum of those of (;/— i)th degree, and so on. Equation 
(3) becomes, when this notation is adopted, 

Pn = O. 

Since this equation is of the ;zth degree it determines n values 
of the ratio of y to x, and each of these values is said to deter- 
mine a point in which the line at infinity cuts the curve. 



Asymptotes. 

228. Applying the above method to the curve traced in 
Art. 212, of which the equation is 



x 3 — y 3 — x 2 + 2y- = o, 



(1) 



we obtain, for the equation P„ — o which determines the points 
at infinity, 

x" —y 3 =o, 
or (x — y) (x~ + xy + j 2 ) = o. 

The second factor gives imagi- 

nary values of the ratio - ; hence the 

x 

curve has but one real point at 
infinity ; namely, that for which 




Fig. 20. 



(2) 



To determine the asymptote the equation of the curve may 
be put in the form 



254 CURVE TRACING. [Art. 228. 



2— — I 

y — x — -5-- — = — 2 - • • • v3; 

' ** + ^/+/ 1 y + y 

x x 2 



If, in this equation, we suppose x and y to become infinite, 

and substitute the value of - from equation (2), we obtain \ 

for the limiting value of the second member ; hence, when the 
generating point recedes to infinity, the value of the first mem- 
ber, y — x, approaches indefinitely to the fixed quantity \. If 
therefore we put 

y-* = b (4) 

we have the equation of a straight line to which the point ap- 
proaches indefinitely ; in other words, this line is an asymptote. 

229. If the given equation be of the ;zth degree, terms lower 
in degree than n — 1 will not in general affect the position of 
the asymptote. Thus, if we have 

x" — xy* 4- ay 1 —a*y = 0, (1) 

the points at infinity are determined by 

x (x + y) (x— y) = o\ (2) 

to find the asymptote corresponding to the factor x — y, we 
put the equation in the form 

**y _ a f 

ay^-af_ * *» . . . . (2) 

y ~ x{x+y) i+ y_ {3) 



§ XXVIII.] ASYMPTOTES. 2$5 



Evaluating this expression when x = y — go, we have for the 
equation of the asymptote 

x-y=-ha (4) 

The term a*y, which is lower in degree than x 2 , does not 
affect the above result, since the corresponding term in the last 
member of equation (3) disappears when x is infinite, and, in 
general, it is evident from the process employed that, when the 
factor which determines the asymptote occurs but once in the ex- 
pression P, ly the position of the asymptote is dependent only upon 
the terms P H -f P n -v 



The Points at which an Asymptote cuts the Curve. 

230. A straight line parallel to an asymptote is said to meet 
the curve at infinity ; since it is evident that when a straight 
line cutting a branch which has an asymptote becomes parallel 
to the asymptote, one of its intersections with the curve must 
disappear by passing to infinity. Accordingly, if we eliminate 
one of the variables by combining the equation of a curve of 
the «th degree with the equation of a line parallel to an asymp- 
tote, the degree of the resulting equation will be denoted by 
n — I. Now, if the line after becoming parallel to the asymp- 
tote be moved into coincidence with it, another intersection 
will recede to an infinite distance. Hence the asymptote is said 
to meet the curve in two coincident points at infinity, or in other 
words to be a ta?igent at infinity ; in fact, it occupies the limiting 
position of a tangent line, when the point of contact with the 
curve passes to infinity. The degree of the equation which 
results from the elimination of one of the variables between the 
equation of the curve and that of the asymptote will accordingly 
be denoted by n — 2 ; and this is the number of finite points in 
which the asymptote meets the curve. 



256 CURVE TRACING. [Art. 23 1, 



231. This circumstance facilitates the determination of the 
points in which the asymptote intersects the curve. Thus, in 
the case of the example discussed in Art. 229, by combining 
equations (1) and (4) to eliminate x we obtain 

— i <fy - i <? = o ; 
hence y = — \ a, 

and from (4) x — — a. 

These are the coordinates of the single point in which the 
asymptote cuts the curve. 

In a similar manner it may be shown that the asymptote 
corresponding to the factor x + y of equation (2) is 

y = - x — \ a, 

and that this line cuts the curve at the point ( — — ,— — 

V 7 14 



Asymptotes Parallel to the Coordinate Axes. 

232. Whenever x or y is a factor of the terms P„, the cor- 
responding asymptote is of course parallel to one of the coordi- 
nate axes (x = o or y = o). In such cases, the position of the 
asymptote is most readily found by the process employed 
below. 

In the case of the curve discussed in Art. 229, since x is a 
factor of equation (2), the corresponding asymptote is parallel 
to the axis of y (x = o). Arranging the equation of the curve 
with reference to powers of y, we have 

{x — d)f + a*y - x* = o (5) 

For each value of x substituted in this equation we have two 



§ XXVIII.] ASYMPTOTES PARALLEL TO THE AXES. 2$? 

values of y ; hence a line parallel to the axis of y generally cuts 
the curve in two points. If however the value of x substituted 
causes the coefficient of y~ to vanish, the corresponding line will 
have but one point of intersection with the curve, hence this 
line must be an asymptote. 

In the above example the coefficient of y~ is x — a ; the 
equation of the required asymptote is, therefore, 

x — a = o. 

By making x = a in equation (5), we obtain the coordinates 
(a, a) of the point at which this asymptote cuts the curve. 

The above process shows that, whenever the expression P n is 
divisible by one of the variables, the equation of an asymptote is 
found by placing equal to zero the coefficient of the highest power 
of the other variable which appears in the given equation. 



Parabolic Branches. 
233. In the case of the curve constructed in Art. 215 ; viz., 
x* — $axy* + 2ay 3 = o, 
the points at infinity are determined by the equation 

x* = o, 

therefore the only points at infinity are in the direction of the 
axis of y. There is, however, no asymptote parallel to this 
axis, since the coefficient of the highest power of y {2d) is 
independent of x, and therefore does not vanish for any value 
of x. Hence the infinite branches of this curve have no asymp- 
totes. 

An infinite branch of this kind is called a parabolic branch. 



258 CURVE TRACING. [Art. 233. 

because the parabolas furnish the most familiar instances of 
infinite branches without asymptotes. 

234. A curve can have a parabolic branch only when the equa- 
tioii P„ — o has two or more equal roots. For, if we apply the 
method of determining an asymptote given in Art. 228, when 
the branch is parabolic the fraction in the second member must 
evidently take an infinite value when x — 00.. Now the denomi- 
nator of this fraction is of the (11 — i)th degree, and the numera- 
tor cannot be of a higher degree; hence, if each term of the' 
fraction be divided by x'*- 1 , the numerator cannot become in- 
finite when x = 00. It follows that the value of the fraction can 

become infinite only when the substitution of the value of - 

x 

causes the denominator to vanish : but, in order that this may 
be the case, it is necessary that the factor of P n , which deter- 
mines the value of — employed, shall also be a factor of the 
x 

denominator ; in other words this factor must occur at least 
twice in P„. 

On the other hand, if P n contains the square of a factor 
not contained in P»_ z , the value of the fraction must be infinite 
for infinite values of x, and the branches corresponding to this 
double root must be parabolic. Thus, if the equation of the 
curve is 

(x - 2j/) 2 {x + y)-a(x* +/) + >■' = o, 

the branches corresponding to (x — 2y) 2 — o are parabolic, as 
may be seen by writing the equation in the form 

x — 21/ = A r^ x » 

[x+-y){x- 2y) 

and putting x = 2y — 00. 



§ XXVIII.] PARABOLIC BRANCHES. 2$g 



A double root of the equation P m = o is not generally to be 
interpreted as indicating the existence of a double point at in- 
finity ; it only indicates that the line at infinity meets the curve 
in two coincident points. In the case of a parabolic branch, 
this line is regarded as a tangent to the curve ; for it is evident 
that as the point of contact of a tangent line recedes indefi- 
nitely on a parabolic branch, the tangent recedes indefinitely, 
instead of approaching a limiting position as in the case of a 
branch having an asymptote. 



Parallel Asymptotes. 

235. We have now to consider the case in which the square 
of a factor appears in P,„ and the first power of the same factor 
in Pn-L For example, let the given equation be 

2x (x — yf — ia {x- — f) + 4<2 2 7 — Ja z = o. . . (i) 

This equation may be put in the form 

, V2 %a (x + y) r \ , 2a*y 7a z 

(x — yf — *—+ ^J- (x —y) + — - — — = o. . 



2x x 2x 



(2) 



Equation (2) may be regarded as a quadratic for determining 
the value of x — y, when x is infinite. 

Putting y = x = 00 in the coefficients, the equation reduces 
to 

(x — yf — la (x - y) + 2cr = o ; 

whence we have 

x— y — 2a and x — y = a, 

the equations of two parallel asymptotes. It will be observed 



260 CURVE TRACING. [Art. 235 

that in finding the equations of a pair of parallel asymptotes it 
is necessary to employ the terms included in the expression 



236. When a curve has a pair of parallel asymptotes, it may 
be shown, by the method employed in Art. 230, that every 
straight line parallel to them meets the curve in two coincident 
points at infinity. Accordingly, if we eliminate one of the vari- 
ables by combining the equation of such a line with the equa- 
tion of the given curve, the degree of the resulting equation 
will be denoted by n — 2. This circumstance facilitates the 
determination of the intersections of the line with the curve, 
and the expressions for the coordinates of the point or points of 
intersection may frequently be used in tracing the curve, as in 
the following example. 

Example 7. x ( y + xf + ay — o (1) 

The method employed in the preceding article gives the 
equations of two parallel asymptotes ; viz., 

y + x = a and y + x = — a. 
The curve has evidently a third asymptote 

x = o. 
The point in which the straight line 

y + X = /3 
meets the curve is determined by the equations, 

* = a^rp and y=aT=f^ 



§ XXVIIL] 



PA RA LLEL A S ) MP TO TES. 



261 




Fig. 23. 



if now we make fl a variable, this point will describe the 
curve. 

Beginning with /S = o, we have x = 
and y ~0\ P is therefore at the origin. 

As ft increases from zero, x is positive 
and j/ is negative ; each numerically in- 
creases until ft = a, when both are infinite, 
P describing the infinite branch in the 
fourth quadrant tangent to the axis of x at 
the origin. See Fig. 23. 

As ft increases from the value #, x is 
negative and numerically decreases to zero, 
the branch described by P approaching the 
axis of y as an asymptote. 

Again, since the given equation is unaltered by changing x 
to — x and y to — y, it is evident that the curve must be sym- 
metrical in opposite quadrants. 

The axis of x is tangent to the curve at a point of inflexion. 
See Art. 219. 

To find the maximum and the minimum ordinate, we derive 
from the value of y in terms of (3, 

dy f?(fh -$<?) , 

dp W-fff ' 

this derivative changes sign only when 

whence (3 = ± a V3. 

These values substituted in the expressions for x and y deter- 
mine the points 

(-&V?>,§aV$) and (W3, - f^3) ; 

the former having a minimum and the latter a maximum ordi- 
nate. 



260 CURVE TRACING. [Art. 235 

that in finding the equations of a pair of parallel asymptotes it 
is necessary to employ the terms included in the expression 
P + P n . z +P n -„ 



236. When a curve has a pair of parallel asymptotes, it may 
be shown, by the method employed in Art. 230, that every 
straight line parallel to them meets the curve in two coincident 
points at infinity. Accordingly, if we eliminate one of the vari- 
ables by combining the equation of such a line with the equa- 
tion of the given curve, the degree of the resulting equation 
will be denoted by n — 2. This circumstance facilitates the 
determination of the intersections of the line with the curve, 
and the expressions for the coordinates of the point or points of 
intersection may frequently be used in tracing the curve, as in 
the following example. 

Example 7. x ( y + xf + cfiy = o ( 1 ) 

The method employed in the preceding article gives the 
equations of two parallel asymptotes ; viz., 

y + x = a and y + x = —a. 

The curve has evidently a third asymptote 

x = o. 

The point in which the straight line 

y + x = /3 

meets the curve is determined by the equations, 

* = ;rr^ and *=2!-?. 1 



XXVI I L] 



PA RA LLEL A S 1 'MP TO TES. 



26l 



if now we make ft a variable, this point will describe the 
curve. 

Beginning with ft = o, we have x = o 
and y = o ; P is therefore at the origin. 

As ft increases from zero, x is positive 
and y is negative ; each numerically in- 
creases until ft = a, when both are infinite, 
P describing the infinite branch in the 
fourth quadrant tangent to the axis of x at 
the origin. See Fig. 23. 

As ft increases from the value a, x is 
negative and numerically decreases to zero, 
the branch described by P approaching the 
axis of y as an asymptote. 

Again, since the given equation is unaltered by changing x 
to — x and y to — y, it is evident that the curve must be sym- 
metrical in opposite quadrants. 

The axis of x is tangent to the curve at a point of inflexion. 
See Art. 219. 

To find the maximum and the minimum ordinate, we derive 
from the value of y in terms of ft, 




Fig. 23. 



dy _ /3 2 Q- 3a 2 ) t 
dp ~ (a* - ft J ; 

this derivative changes sign only when 



ft" - 3* s 



o, 



whence 



ft= ±aV$. 



These values substituted in the expressions for x and y deter- 
mine the points 

(-*»V3»f»V3) and (W3>-W3); 

the former having a minimum and the latter a maximum ordi- 
nate. 



264 CURVE TRACING. [Art. 239. 

powers of (x — y), it is necessary to divide by x to obtain terms 
which have a finite value when x is made infinite ; this causes 
the first term to vanish, and gives the simple equation 

— a (x — y) + a 1 = o, 
hence x— y — a = o 

is the equation of an asymptote. 

The remaining infinite branches of the curve are parabolic, 
hence the line at infinity touches the curve at the point at 
which the asymptote also touches it ; the curve is therefore 
said to have a double point at infinity at which the line at infinity 
is one of the tangents. 

This curve is a trident ; since the characteristic singularity 
of a trident is a node of the kind described above. Compare 
Art. 209. The curve may be traced by the method employed 
in Art. 236. 



240. If the equation employed in determining parallel 
asymptotes has two equal roots, the curve has two coincident 
asymptotes, and, in general, two real branches approaching the 
same part of the common asymptote on opposite sides and 
forming a cusp at infinity. See Fig. 17, Art. 202. 

The corresponding branches may, however, in a case of this 
kind be imaginary, as in the following example : 

xy + a V - a 6 = o. 

Each coordinate axis presents a case of coincident asymptotes. 
Solving the equation, we obtain 



§ XXVIII.] 



NODES AT INFINITY. 



265 



y — o when x — ± a\ as 
x numerically decreases, 
both values of y increase 
indefinitely, giving the 
branches represented by 
the full lines in Fig. 24. 
When x numerically ex- 
ceeds #, y is imaginary; 
hence there are no real 
branches approaching 
the axis of x. If how- 
ever we change the sign 
of the term containing 
y*- in the given equation, 
we shall have the curve 



Fig. 24. 



y 



±"-V(x*-a% 



represented by the dotted branches in the figure ; the real 
branches of this curve being those to which the axis of x is an 
asymptote. 

Examples XXVIII. 

Find the asymptotes of the following curves. 

t. (x + a)y* = (y + b) x*. x = — a, y = — b, y = x + b — a. 

2. x s — ^xy 1 — 3JC 2 + i2xy — \2y l + Sx + 2y + 4 = o. 

x = — 3, x = 2y f x + 2y = 6. 

3. ( y — 2x) (/ — # 2 ) — a (jy — xY + 4a 2 (# -f y) — a s — o. 

j = #, j; + # = fa, _y = 2X + £0. 

4. ^y + tf* (# -f j') 2 — 2a i y i — a' — o. a: = — 20, x — a. 



266 CURVE TRACING. [Ex. XXVIII. 

5. x i y i — (x 2 — y) 2 + y — 1 =0. x = ± i,y = ± 1. 

6. ( y — x) (x 2 — a 2 ) = d\ y = x, x = ± #. 

7 3 4 1 4 3 2 4 

7. # — #j/ + # jy — tf,Xjy = o. 

x = o, x — — a, x -V y — \a, x —y = \a. 

8. x 2 (x —y) 2 — d l (x 2 + y) = o. x = ± a, y — x ± a^/2. 

9. (x 2 — y 2 ) 2 — 4/ + y — o. x—y — ± 1, x + y = ± 1. 

10. Trace the curve x 3 + y % — x 2 — y 1 = o. 

The origin is an acnode. Since x and jy are interchangeable in this 
equation, the curve is symmetrical with reference to the line y = x ; 
find the point at which the curve cuts this line, and show that the 
asymptote x +y = § does not cut the curve. 

11. Trace the curve x b + y b — $ax*y = o. 

The asymptote is y 4- x = —a, cutting the curve in the second 
quadrant. The point (a fy 162, a fy 108), has a maximum ordinate, 
and the point (2a ^/ 8, 2a ^/ 2) has a maximum abscissa. 

12. Trace the curve y* + 2axy 2 — x* = o. 

Find the two asymptotes, and show that each intersects the curve at 

points corresponding tox= ± - ^2. 

13. Trace the curve x b — 2a s xy + y b = o. 

The asymptote passes through the origin ; the curve is symmetrical 
with reference to the line y ~ x, and cuts it at the point (a, a). 

14. Trace the curve x b — ^ax^y 1 + y h = o. 
The equation of the asymptote isy + x = a. 

15. Trace the curve x* — y s + (27 — x) 2 = o. 

Determine the direction of the curve at the points in which it cuts 
the coordinate axes, and at the point determined by m = — 1. The 
curve has a cusp at the origin. 



8 XXVIII.] EXAMPLES, 267 



16. Trace the curve x*y* — x*y + x* —±f — o. 

The four asymptotes are x = ± 2, j = x, y = o. 

17. Trace the curve x 2 y — y*x + a: 2 — 4_y 2 = o. 

Find the asymptotes, the points at which they cut the curve and 
the direction of the curve at these points. 

18. Trace the curve xy 2 + x 2 y + 2x 2 — $xy — 2_y 2 = o. 

Find the three asymptotes, and the point in which each cuts the 
curve. 

19. Trace the curve 4X* — ^xf + ay 2 — $ax 2 — o. 

Show that the three asymptotes pass through a common point. 

20. Trace the curve (x + 2y) (x — y) 2 — 6a 2 (x -{- y) = o. 

The curve is symmetrical in opposite quadrants, and has three 
asymptotes two of which are parallel. 

si. Trace the curve y (y — xf (y + 2x) — 3^'V. 
The curve is symmetrical in opposite quadrants, and has four 
asymptotes two of which are parallel. 

22. Trace the curve y* — y % x + x % — 2x 2 y = o. 

The curve has parabolic branches, and an asymptote that cuts the 
curve in two points. 

23. Trace the curve x i — ax^y — axy 2 + a 2 y 2 = o. 

The origin is an isolated point (see Art. 224). The curve has 
parabolic branches, and the line x — a is an asymptote that cuts the 
curve in a single point. The line y = mx touches the curve when 
m = 1 and when m = — 3. By solving for jy, it may be shown that the 
line x = \a touches the curve. 

24. Trace the curve x i — ax 2 y 4- axy 2, + \a 2 y 2 = o. 

The equation of the asymptote is x = — \a. The curve has para- 
bolic branches and a ramphoid cusp at the origin. 

25. Trace the curve x k — \ax 2 y — axy 2 + a*y* — o. 



268 CURVE TRACING. [Ex. XXVIII. 

The line y = mx touches the curve when m = — -|, and when 
m = — |. Show that both branches of the curve pass through the 
origin, and find the asymptote and the point at which it cuts the 
curve. 

26. Trace the curve x h — 4ay A + 2ax*y + cfxy 1 = o. 

The origin is a triple point at which two of the branches form a 
ramphoid cusp. The line y = x touches the curve at (a, a). The 
infinite branches are parabolic. 

27. Trace the curve (x —y)* + a (x^ — y*) + a*y = o. 

This curve is a trident touching the axis of x at the origin. See 
Art. 239. 



XXIX. 

Curves Given by Polar Equations. 

241. The following examples will illustrate some of the 
methods employed when the curve is given by means of its 
polar equation. 

Example 8. r = # cos #cos 20 (1) 

When 6 — o, r — a, the generating point P therefore starts 
from A on the initial line. As 6 increases, r decreases and 
becomes zero when 8 = 45 , P describing the half-loop in the 
first quadrant, and arriving at the pole in a direction having an 
inclination of 45 to the initial line. When 6 passes 45 , r 
becomes negative, and returns to zero again when 6 = 90 , P 
describing the loop in the third quad- 
rant. As 6 passes 90 , r again becomes 
positive, but returns to zero when 
6= 135 , P describing the loop in the 
second quadrant. As Ovaries from 135 
to 180 , r again becomes negative, P de- 
FlG ' 25 ' scribing the half-loop in the fourth quad- 

rant, and returning to A. 




§ XXIX.] CI T R VES GIVEN B Y POL A R EQ I 'A TIONS. 269 

In this example if we suppose 6 to vary from 180° to 360 , 
P will again describe the same curve, and, since 6 enters the 
equation of the curve, by means of trigonometrical functions 
only, it is unnecessary to consider values of 6 greater than 360 . 

242. Putting equation (1) in the form 

r = a (2 cos 3 — cos 0), 



we derive 



dr 

— = a(— 6 cos-6 sin 6 + sin 6). 
do 



To determine the maxima values of r, we place this derivative 
equal to zero, thus obtaining the roots 

sin 6~o and cos 6 = ± J V6 ; 

the former gives the point A on the initial line, and the latter 
gives the values of 6 which determine the position of the maxi- 
ma in the small loops. The corresponding values of r are =F - V6. 

To determine the position of the maximum ordinate, we 
have from (1) 

y = r sin 6 = £ a sin 46. 

The maxima values occur when sin 4.0=1, and the minima 
when sin 46 = — 1 ; that is, we have maxima when B — \n and 
when 6 = \n, and minima when O—^n and \n. 

243. In the preceding example the substitution of 6 + n 
for 6 changes the sign but not the numerical value of r. When 
this is the case, #and 6 +tt evidently give the same point of 
the curve, and the complete curve is described while 6 varies 
from o to 7t. If however this substitution changes neither the 
numerical value nor the sign of r, 6 and 6 + n will give points 
symmetrically situated with reference to the pole ; that is, the 
curve will be symmetrical in opposite quadrants. 



270 CURVE TRACING. [Art. 243. 

Again if the substitution of — 6 for 6 does not change the 
value of r, 6 and — 6 give points symmetrically situated with 
reference to the initial line, hence in this case the curve is sym- 
metrical to this line ; but, if the substitution of — 6 for 
changes the sign of r without changing its numerical value, the 
curve is symmetrical with reference to a perpendicular to the 
initial line. 



The Determination of Asymptotes by Means of Polar 

Equations. 

244. When r becomes infinite for a particular value of 6 the 
curve has an infinite branch, and, if there be a corresponding 
asymptote, it may be determined by means of the expression 
derived below. 

Let 6 l denote a value of 6 for which r is infinite, and let OB 
be drawn through the pole, making this angle with 
the initial line ; then, from the triangle OBP, Fig. 
'? 26, we have 

I PB=rsm(d l -ff). 

Now, if the curve has an asymptote parallel to 
OB, it is plain that as 6 approaches 1 the limiting value of PB 
will be equal to OR, the perpendicular from the pole upon 
the asymptote. Hence, if the curve has an asymptote in the 
direction 6 lf the expression 

OR =[r sin (6, - 6)\, 

which takes the form 00 ■ o, will have a finite value, and this 
value will determine the distance of the asymptote from the 
pole. Fig. 26 shows that when the above expression is posi- 
tive OR is to be laid off in the direction 6 I — 90 . 

If upon evaluation the expression for OR is found to be in- 




XXIX.] 



ASYMPTOTES. 



27I 



finite we infer that the infinite branch of the curve is para- 
bolic. 



245. Example 9. 



r = 



a6* 



s 



(0 



Since r becomes infinite when 6 = 1, we proceed to apply the 
method established in the preceding article for determining the 
existence of an asymptote. In this case we have 



[rsin^-0)] 



ad 2 



6 



sin(i - ey 



6 



= — ha. 



The angle = 1 corresponds to 57 18', nearly, and since 
the expression for the perpendicular on the asymptote is neg- 
ative its direction is 6 X + 90 = 147 18'; consequently, the 
asymptote is laid off as in Fig. 27. 

Numerically equal positive and negative values of 6 give the 
same values for r ; hence the curve is symmetrical with refer- 
ence to the initial line. 

While 6 varies from o to 1, r is negative and varies from O 
to co, giving the infinite branch in the third 
quadrant. 

As passes the value unity, and increases 
indefinitely, r becomes positive and decreases, 
approaching indefinitely to the limiting value 
a y which we obtain from (1) by making 6 in- 
finite. Hence the curve describes an infinite 
number of whorls approaching indefinitely to 
the circle r = a, which is therefore called an / 
asymptotic circle. 

The points of inflexion in this curve are 
determined in Art. 325. 




Fig. 27. 



272 CURVE TRACING. [Ex. XXIX. 



Examples XXIX. 

i. Trace the curve r = a cos 3 \ 0. 

Show that, to describe the curve, must vary from o to 37? ; also 
that the curve is symmetrical to the initial line. Find the values of 6 
which correspond to the maxima and minima ordinates and abscissas, 
the initial line being taken as the axis of x. 

2. Trace the curve r — a (2 sin — 3 sin 3 0). 

Show that the entire curve is described while varies from o to 7t, 
and that the curve is symmetrical with reference to a perpendicular 
to the initial line. 

3. Trace the curve r = 2 + sin 30. 

A maximum value of r (equal to 3) occurs at = 30 ; a mini- 
mum (equal to 1) at = 90 . The curve is symmetrical with refer- 
ence to lines inclined at the angles 30 , 90 , and 150 to the initial 
line. 

4. Trace the curve r = 1 + sin 50. 
The curve consists of five equal loops. 

5. Trace the curve r 3 = a 5 sin 30. 

The curve consists of three equal loops. 

6. Trace the curve r cos = a cos 20. 

The curve has an asymptote perpendicular to the initial line at the 
distance a on the left of the pole. 

7. Trace the curve r — 2 -h sin |0. 

A maximum value of r occurs at = 6o°, and a minimum at 
= 180 . The curve has three double points, one being on the initial 
line. 

8. Trace the curve r cos 20 = a. 

The curve is symmetrical with reference to the initial line and 
with reference to a perpendicular to the initial line. There are four 
asymptotes. 



§ XXIX.] EXAMPLES. . 273 

9. Trace the curve ;-sin 40 = a sin 30. 

The curve is symmetrical to the initial line, and has three asymp- 
totes ; the minimum value of r is \a. 

10. Trace the curve r a = a 1 cos 20. 

The curve is symmetrical with respect to the pole since 
r = ± rt 4/ (cos 20) : r is imaginary for values between \n and \n. 

11. Trace the curve r A = #* cos Jo. 

The curve consists of three equal loops, r being real for all values 
of e. 

12. Trace the curve r = ^ (cos 4- cos 26). 

The curve consists of three loops with a cusp at the origin. De- 
termine the two double points by the condition that and +180 
shall give the same point of the curve. 

6. 5. 

13. Trace the curve r :i — a* cos f 3. 
The curve consists of five equal loops. 

14. Trace the curve r~ sin = a 2 cos 20. 

The curve consists of two loops and two infinite branches, to which 
the initial line is a common asymptote. 

15. Trace the curve r 2 cos = a 2 sin 30. 

The curve consists of two loops and an infinite branch which has 
an asymptote perpendicular to the initial line and passing through the 
pole. 

16. Trace the curve r sin = a sin (20 + a). 

Show that and S + n give the same point of the curve. The 
curve cuts the asymptote at the point = \n — a, r = a tan a. 

17. Trace the curve r — a (tan — 1). 

Show that the curve is symmetrical with reference to the pole as a 
centre, and determine the maximum abscissa. 

18. Trace the curve r = a . 

20 — 1 



274 CURVE TRACING. [Ex. XXIX. 

Find the rectilinear and the circular asymptote, and also the point 
of inflexion. 

19. Trace the curve r = 1 ± 4/(2 — cosec 0). 

Show that, to describe the curve, must vary from o to 27T, and 
that it consists of three branches, one of which is a closed curve, the 
other two having a common asymptote. 

20. Determine the asymptotes of the curve 

r cos mQ = a cos nQ. 

ll J rf+1 i , ;OJ=(-,)''c^ ,(ilf » . 
2#z ^ 2m 

21. Trace the curve r (0 — tt) 2 = a (0 2 — J7r 2 ). 

The infinite branches are parabolic ; the negative values of r cor- 
responding to values of between — \n and \n determine a loop ; the 
curve approaches the asymptotic circle both from within and from 
without. Find the point at which the parabolic branch cuts the 
asymptotic circle. 

22. Trace the curve r = 



+ sin 



The curve is symmetrical to the initial line ; values of numeri- 
cally less than n give a loop enclosing the pole ; for greater values of 
each whorl cuts the asymptotic circle at its intersection with the 
initialTine. 

, m a (0 + cos 0) 

23. Trace the curve r = — ■: . 

+ sin 

The curve has an asymptote parallel to the initial line ; the whorls 
cut the asymptotic circle at the extremities of a diameter" inclined at 
an angle of 45 ° to the initial line. 



§ XXX.] TRANSCENDENTAL CURVES. 275 

XXX. 

Transcendental Curves. 

246. The algebraic curves constructed in the preceding sec- 
tions furnish examples of points of inflexion, nodes, cusps and 
multiple points. The term singular point is used to include 
these and also two varieties of points which occur only in 
curves whose equations involve transcendental functions. The 
following examples of transcendental curves will serve to illus- 
trate these two varieties of singular points ; viz., stop points or 
points <Tarrit at which a curve terminates abruptly, and salient 
points ox points anguleux at which two branches of a curve meet 
without having a common tangent. 



Stop Points. 

1*\n. Example 10. y — g®. 

Since, by Art. 113, this function has the limiting value infinity 
when x is positive and approaches zero, and the limiting value 
zero when x is negative and approaches zero, the right-hand 
branch has the axis of y for an asymptote, 
while the left-hand branch stops at the origin. 

When x is infinite, y = I ; hence the line 
y = 1 is an asymptote to each branch of the 
curve. 

Taking the derivative, we have Fig. 28. 

1^ 

dy _ s * 

dx~~ ~x~ 2f 
which, by Art. 114, is zero when x is negative and approaches 



276 



CURVE TRACING. 



[Art. 247. 



zero. Hence the left-hand branch is tangent to the axis of x, 
and the form of the curve is that represented in Fig. 28. 
Taking the second derivative, 



dx 2 ' 



(2x 4- iy> 



we find a point of inflexion corresponding to x = — \ ; the 
ordinate of this point is £~ 2 = 0.14 ... . 




248. Example 11. 



In this case we evidently have y = o, whether x approaches 
zero from the positive or from the negative side. 

The value of tan <f> at the origin is (see Art. 118) the same 

as that of — but in this case 



-4] 

1 4- «*-i. 



When x is positive and approaches zero, the value of this ex- 
pression is zero, but when x is negative and approaches zero 
its value is unity ; hence the right-hand branch touches the 
axis of x at the origin, while the left-hand branch is inclined at 
an angle of 45 °. See Fig. 29. 

249. To determine the infinite branches of this curve we 
have, from equation (1), 




= *. 



Fig. 29. 



To ascertain whether the curve has an 
asymptote, we put the equation in the form 



§ XXX.] 



SALIENT POIXTS. 



277 



X X I — €* 

y-$x = 1 -i*=- r 



+ f j 



and evaluate for x = 00. 
Putting — = z, we have 



I + € x 



I — €' 



y -W- = WT?)- 



Hence the equation of the asymptote is 

y = & — i- 

Branches Pointillees. 



250. Example 12. 



Whence 



y = x x . 

d £ = ,-{, + log.) 



since by evaluating we have 

7 = I and tan <f> = 00 when ,r = o ; hence the curve touches the 
axis of y at the point (o, 1). It passes through the point (1, 1) 
at an inclination of 45 to the axis of x, and 
the ordinate is a minimum at the point 



[!■©■]■ 



251. When x is negative the function 
x x is not continuous ; it is in fact real only 
when the value of x can be expressed by an 
integer or by a fraction having an odd denominator. When 



278 CURVE TRACING. [Art. 25 1. 



this is the case, the numerical value of each real ordinate is the 
reciprocal of the ordinate corresponding to a numerically equal 
positive value of x \ hence there is an unlimited number of 
points situated on branches having the shape indicated by the 
dotted lines in Fig. 30. 

Branches having this character are called branches pointil- 
lees. 

When j is a positive fraction having, when reduced to its 
lowest terms, an even denominator, the value of x* is obtained 
by extracting a root which admits of the ambiguous sign. If 
the symbol x* is regarded as expressing at once both values of 
the function (that is, if for example when x — J, we write 
x* = ± |/J) the curve y = x x must be regarded as having 
also a brancJie pointille'e in the fourth quadrant, symmetrical 
to the continuous branch. 

Examples XXX. 

1. Trace the curve jy = x log.*. 

Show that there is a stop point at the origin. Find the point at 
which y is a minimum, and the direction of the curve at the origin," at 
the point (1,0), and at the point (e 7 e). 

2. Trace the curve y x = 1 — cos.*. 

The curve has a stop point at the origin ; to find the direction of 

V f 1 — cos X \ — 
the curve at this point, evaluate — — f r for x — o. 

X \ X / 



3. Trace the curve 7 (1 4- s x ) = 1. 

Find the asymptote and the direction of the curve at the two stop 
points. Show, by transferring the origin to the point (o, J), that the 
curve is symmetrical about this point as a centre. 

I 1 

4. Trace the curve y (sx — 1) = x(s x + 1). 

The curve has a salient point at the origin, and is symmetrical 
with reference to the axis of y ; the branches are parabolic. 



§ XXX.] EXAMPLES. 279 

5. Trace the curve y (1 — s m ) =x(i -—**). 

The curve has a salient point at the origin, and an asymptote 
parallel to the axis of x. 

6. Trace the curve y (1 -I- s x ) — x (1 — $e* ). 

There is a salient point at the origin. Find the point in which the 
curve cuts the axis of x, and find also the asymptote. 

7. Trace the curve y — xe~-\ 

Find the point of inflexion, and the point at which the ordinate 
has a maximum value. 

8. Trace the curve y = £ C09Z , and find the points of inflexion. 

9. Trace the curve y*^ — x* 4- 1 = o. 

Find the points at which tan Q is zero ; also those at which tan Q 
is infinite. The axis of x is an asymptote. 

10. Trace the curve y = (1 — x)*. 

When x > 1, we have brcuiches pointille'es. Find the inclination of 
the curve at the points for which x = — 1, x = o, and x = 1. 

11. Trace the curve v = x- . 

The curve has a stop point at the origin, and a branche pointille'e 
in the second quadrant. Determine the maximum ordinate, the incli- 
nation of the curve at the point (1, 1), and also at the origin (the lat- 
ter by the method employed in Art. 248). 

12. Trace the curve v = c~'\ and find the points of inflexion. 
This curve occurs in the Theory of Least Squares and is known as 

the Probability Curve. 

13. Trace the curve y = 6 sin_ '. 

The curve consists of an unlimited number of branches, each con- 
taining a point of inflexion. Find the point of inflexion in the branch 



280 CURVE TRACING. [Ex. XXX. 

corresponding to the primary value of x and the angle at which this 
branch cuts the axis of y. 

14. Trace the curve y = 1 — x. 

The curve has an asymptote, and the portion corresponding to 
values of x greater than unity consists of branches pointillees. Find 
the inclination' of the continuous branch at the points at which it cuts 
the axes. 

15. Trace the curve j> = (sin x) x . 

Find the direction of the curve when x = o and when x = n. The 
curve has alternately continuous branches and branches pointillees. 



CHAPTER IX. 

The Equations and Constructions of Certain Higher 

Plane Curves. 



XXXI. 



252. In this chapter are given the definitions, equations, 
and constructions of certain curves, some of which possess an 
interest chiefly historical, while others, on account of their 
peculiar properties, are of frequent occurrence in works pertain- 
ing to mathematical subjects.** 

The Parabola of the nth Degree. 

253. The term parabola is frequently applied to any curve 
in which one of the coordinates is proportional to the ;/th 
power of the other, n being greater than unity. The parabola 
proper is thus distinguished as the parabola of the second 
degree. , 

The general equation of the parabola of the nth. degree is 
usually written in the homogeneous form, {a being positive) 

a»- T y = x n . 

* A full index to the curves given in this chapter will be found in the table of 
contents. The cuts have been prepared from diagrams accurately constructed in 
accordance with the definitions of the curves discussed. For a number of these 
diagrams we are indebted to Cadet Midshipman J. H. Fillmore. They were first 
published in Part III. of the preliminary edition of this work, printed, for the use 
of the cadets at the U. S. Naval Academy, at the Government Printing Office, 
Washington, D. C, 1876. 



282 



CERTAIN HIGHER PLANE CURVES. [Art. 2 $3. 



The curve passes through the origin and through the point 
(a, a), for all values of n. Since n > 1, the curve is tangent to 
the axis of x at the origin. 

254. The following three diagrams represent forms which 
the curve takes for different values of n. When n denotes a 
fraction, it is supposed to be reduced to its lowest terms. 

Fig. 31 represents the general shape of 
the curve when n is an even integer, or a 
fraction having an even numerator and an 
odd denominator. 

Fig. 32 represents the form of the curve 
when n is an odd integer or a fraction with 
an odd numerator and an odd denominator, 




Fig. 31. 



the origin being a point of inflexion. 

Fig. 33 represents the form of the 
curve when n is a fraction having an odd 

an even denominator. 

is regarded as a two- 



numerator and 

In this case y 

valued function, and is imaginary when 

x is negative. 




Fig. 32. 

Fig. 31 is constructed for the parabola 
in which n = 4. 

Fig. 32 is the cubical parabola in which 
n= 3. 

Fig. 33 is the semi-cubical parabola in 
which #=={■'; the equation being 



a*y — ± x* y 



Fig. 33. 



or 



ay 1 



x . 



The curves corresponding to the general equation 
y = A + Bx + Cx* + Dx 3 + . . . Lx tl 
are sometimes called parabolic curves of the #th degree. 



I XXXI.] 



THE CISSOID OF DIOCLES. 



283 



Th c Cissoid of Diodes. 

255. Let A be a point on the circumference of a 
and BC a tangent at the opposite 
extremity of the diameter AB\ let 
AC be any straight line through A, 
and take CP = AD\ then the locus 
of P is the cissoid. 

To find the polar equation, AB 
being the initial line, let DB be 
drawn, and denote the radius of the 
circle by a; then AC = 2a seed; 
and since ADB is a right angle, 
AD — 2a cos 6. The polar equation 
of the locus of P, A being the pole, 
is, therefore, 




r — 2a (sec 6— cos 6) = 2a 



cos- 



Fig. 34. 

e 



cos 6 



or 



r = 2a 



sin' 



cos 



(1) 



256. To obtain the rectangular equation, we employ the 
equations of transformation 



sin 6 



cos 6 



r« = ** + /; 



whence, eliminating we obtain 



2a 



rx 



and thence the rectangular equation of the curve 

x (x°- + f) = 2ay\ . . . 
x z 



or 



/= 



2a 



(2) 
(3) 



284 



CERTAIN HIGHER PLANE CURVES. [Art. 257. 



The Cissoid Applied to the Duplication of the Cube. 

257. The Greek mathematician Diodes, who is supposed 
to have lived in the sixth century A.D., was the inventor of 
this curve, which was employed by him in constructing the 
solution of the problem of finding two mean proportionals, of 
which the duplication of the cube is a particular case. 

In Fig. 35, OE is a radius per- 
pendicular to AB, and OS is a 
given length denoted by b ; let BS 
intersect the curve in P, and let 
AP intersect OE in T; then, de- 
noting OT by z } we have, by simi- 
lar triangles, 




2a — x 



and 



By combining these equations with 
the rectangular equation of the cis- 
soid, we deduce 



Fig. 35. z=J/(a'b). 

The above construction may be used to find the first of two 
mean proportionals between two given quantities a and b ; for, 
if we insert two geometrical means between a and b y we have 

In the duplication of the cube, we have b = 2a, therefore 

z — ay 2, or z* = 2a\ 

Example. 

1. Show, by means of the rectangular equation of the cissoid, that 
it has a cusp at the origin, and an asymptote ; also find the inclination 
at the point {a, a). 



>< XXXI.] THE COX CHO ID OF NICOMEDES. 



28 5 



The Conchoid of rVzcomedes. 

258. Let A be a given point and BC a given straight line. 
On any line A C take CP, and also CP\ 
equal to a given constant quantity 
denoted by b ; the locus of P and P' 
is the conchoid. 

A is the pole, BC the directrix, and 
b the parameter. 

The locus of P is called the infe- 
rior branch, and that of P' the supe- 
rior branch. 

Denoting AB by a, we have, for 
the polar equation, 



r = a sec 6 ± b. . 



(1) 



259. To obtain the rectangular 
equation, we put 




Fig. 36. 




Fig. 37. 



sec = -, and r = V {**+/) ; 
whence 

^O 2 + f) (* - a ) = ± bx, (2) 
or (x-+/)(x-ay = frx\ (3) 

Equation (3), being equivalent to 
both equations (2), represents both 
branches of the curve. 

When b > a, the inferior branch 
of this curve has a double point at 
A and a loop on the left, as shown in 

Fig. 37- 

When b — a, the point A becomes 
a cusp. 



286 



CERTAIN HIGHER PLANE CURVES.' [Art. 259. 



Nicomedes, the inventor of this curve, was a Greek mathe- 
matician of the second century A.D. 

The Trisection of an Angle by means of the Conchoid. 

260. The conchoid, like the cissoid, was employed in con- 
structing the solution of the problem of two mean proportion- 
als ; it was also applied to the trisection of an angle, another 
celebrated problem of the ancient geometers. 

Let CAB be the given acute 
angle, and let CB be drawn per- 
pendicular to AB. Construct the 
conchoid of which A is the pole, 
BC the directrix, and 2AC the 
parameter. Let CP parallel to 
AB intersect the curve in P; then 
PAB is \ of CAB; for, denoting 
CAP by <p,we have 




sin^ __ sin 



e 



Fig. 38. 



whence 
therefore 



bcos 6 

sin cp — 2 sin 8 cos 6 = sin 26 ; 

e = ^cp = i CAB. 

Example. 



1. Find, from the equation of the conchoid, the tangents at the 
origin, and thence determine when the curve has a crunode, when an 
acnode, and when a cusp. 

The Quadratrzx of Dinostratus. 

261. Let the radius of the circle in Fig. 39 revolve uni- 
formly, completing the semicircle AEB in the same time as 



§ XXXI.] THE QUADRATRIX OF DINOSTRATUS. 



287 



that required by the ordinate RP to move uniformly over the 
diameter; the intersection P of the ordinate and the radius 
will then describe the curve known as the quadratrix. 




Fig. 39. 

Denoting the radius by a, the angle at the centre by 6, and 
taking the origin at A, we have, by the mode of construction, 



- = .£. 

7t ~ 2a ' 

Eliminating 6, we obtain 



and y = (a — x) tan 0, 



j, = (a-x)tan—, 

the equation of the quadratrix. 

262. It is evident that, if AR be divided into any number 
of equal parts, and ordinates be erected at the points thus 
determined, the corresponding radii of the circle will divide the 
angle A CP into the same number of equal parts. Hence, by 
means of this curve, an angle rflay be divided into any number 
of equal parts. The curve was employed for this purpose by 
Dinostratus, a disciple of Plato ; he also employed it in the 
quadrature of the circle. The latter application, from which 



288 



CERTAIN HIGHER PLANE CURVES. [Art. 262. 



was derived the name of the curve, depends upon the result 
deduced below. 

By evaluation, we obtain 



CD = (a — x) tan 



nx 

2a 



2a 

7t 



hence we have 



AB 
CD 



= 7t. 



The Witch of A guest. 

263. Given a point A on the circumference of a circle, and 

a tangent at the opposite extremi- 
ty of the diameter AB ; if the or- 
dinate DR be produced, so that 
PR = BC, the locus of P will be the 
witch. 

Taking the origin at A, and 
denoting the radius of the circle 
by a, we have 

DR t = x{2a — x)\ 




also, by the construction of the 
locus, 



Fig. 40. 



2a 



DR 

x 



therefore 



xj? = 4a* (2a — x) 



is the rectangular equation of tie curve. 

This curve was given under the name of versiera in a treatise 
on Analytical Geometry by Donna Maria Agnesi, an Italian 
mathematician of the eighteenth century. 



§ XXXI.] 



EXAMPLES. 



Examples. 

1. Show, by means of the equation of the witch, that the axis of;' 
is an asymptote, and that the curve has two imaginary asymptotes 
parallel to the axis of x. 

2. Find the points of inflexion in the witch, and the inclination of 
the curve at these points. 



CI* ± !*.):■£) = * J*. 



The Folium of Descartes, 

264. This name has been given to a cubic defined by the 
equation 



x + y — ^axy — o. 

The form of this curve is represented in 
Fig. 41; the axes are tangent to the curve at 
the origin. By the method employed in Art. 
228, the equation of the asymptote is found 
to be 

x + y + a = o. 




Examples. 

1. Show that the asymptote does not meet the curve, and find the 
point at which the ordinate is a maximum. { a <\/ 2 , a ^/4)- 

2. Show that, when the axes are turned through an angle of 45 °, 
the equation of the folium is 

x 3 + 3xy 2 — I Y2 . a (x*— y 2 ) = o. 

By means of this equation, find the asymptote and the point at which 
the ordinate is a maximum. 

liV6a, §^ 3 (V3-i)«]. 



290 



CERTAIN HIGHER PLANE CURVES. [Art. 265, 



The Strophoid or Logocyclic Curve. 

265. Let AB be perpendicular to the line BC, and denote 
its length by a. Let AC be any straight 
line through A, and take CP and CP each 
equal to CB ; then the locus of /^ and P' 
will be a continuous curve called the 
strophoid. 

Since AC— a sec 6, and (7i? = a tan (9, 
the polar equation of the curve is 




r = a (sec 6 ± tan 6). 



(1) 



As (9 approaches 90 , P approaches 

indefinitely near to A, while P' describes 

the infinite branch BP' . Since the per- 

IG ' 42 ' pendicular distance of F from the line 

BC is the same as that of P, it is evident that the curve- will 

have an asymptote parallel to BC at the distance a from it on 

the right. 

This curve was styled the logocyclic curve by Dr. Booth, 
who discussed its properties elaborately in a paper read before 
the Royal Society, June 10, 1858. 

266. To derive the rectangular equation of the curve, we 
substitute in equation (1) 



sec 



6 = 



and tan 6 



I- 
x 



whence 



\x xJ 



or 



(-2- 



x ' 



§ XXXI.] THE STROP HOID OR LOGOCYCLIC CURVE. 2g\ 



Squaring, (*■ + .r)( l - -J = -pr » 

reducing (;tr — ay + j • — — -- = o ; 

x 

x(x - (if + J' 2 (> — 2tf) = o (2) 

This is the rectangular equation of the strophoid referred to A, 
Fig. 42, as the origin. 

267. Transferring the origin to the point B {a, o), we have 

x\x + a) -f y(:r — «) = o, 

or x(x> + f) + tf(;r 2 - f) = o ; (3) 

whence it is evident that the tangents to the curve at the node 
are the lines 

y = x and y = — x. 

From (3), by passing to polar coordinates, we derive the 
equation 

r s cos 6 + ar* (cos 2 6 — sin 2 6) = o, 

the factor r 2 indicating the node at the pole B. Rejecting this 
factor, we have 

r cos 6 + a cos 20 — o. 

Reversing the direction of the initial line by putting 6 + 180 
in place of 6, we have 



292 



CERTAIN HIGHER PLANE CURVES. [Art. 267. 



r = a 



cos 26 
cos 6 



(4) 



the polar equation of the strophoid, B being the pole and BA 
the initial line. 

Examples. 

1. Find the position of the maximum ordinate of the strophoid. 

The origin being A, x = \ (3 — 4/5) a. 

2. Let A be sl fixed point in the circumference of a circle, and AB 
any chord passing through it ; draw a diameter parallel to AB, and 
through B a line parallel to the tangent at A ; prove that the locus of 
the intersection of this line with the diameter is the strophoid. 



The Limacon of Pascal. 

268. If through a given point A on the circumference of a 
circle a line be drawn cutting the circumference again at C, and 
from the latter point a given distance be laid off in each direc- 
tion on this line, the locus of the points thus determined is 
called the limagon. 

Let the diameter of the circle 
ACB, Fig. 43, be denoted by 2a, 
and the given constant by b ; 
then the polar equation of the 
locus of P and P' will be 




r = 2a cos 6 ± b. 



(1) 



Fig. 43. 
obtain the point defined by 



It is to be observed that each 
of these equations gives the en- 
tire curve; for, if we put /2+180 
for 6, and use the lower sign, we 



§XXXI.| 



THE LIMACON OF PASCAL. 



293 



6 = a + 180 , and r = — 2a cos a — b ; 



but this is the same as the 
point determined by 

6 = a, and r — 2a cos <r + b, 

I 
the latter being obtained by '; 

using the upper sign in equa- 
tion (1). 

Reversing the direction of 
the initial line, we have 

r = b — 2a cos 6, . (2) 

another form of the polar 
equation. 




Fig. 44. 



269. Transforming (1) to rectangular coordinates, we have 



r — 2a - ± b ; 
r 



whence x* + y 2 — 2rtur — ± b 1/(V + y), 

(^-f/) 2 -( 4 ^+^)(^+/)+4^ 2 = o. . . . (3) 



or 



When b > 2a y the curve takes the form indicated in Fig. 44. 

The limacon occurs as a particular case of the Cartesian 
ovals (see Art. 275, Ex. 3) ; of the epitrochoid (see Art. 296) ; and 
of the hypotrochoid (see Art. 299). 



270. The limacon in which b — a has been used in the tri- 
section of an angle, and is hence called the trisectrix. Its. polar 
equation is 



2 9 4 



CERTAIN HIGHER PLANE CURVES. [Art. 270. 



r = a (2 cos 6 ± 1), 
and putting b = a in (3), we have for its rectangular equation 
(V + /)■ - 4ax(x* + f) + 30V - tf 2 / = o. 



Examples. 

1. In the case of the limaeon, express x in terms of 0, and thence 
determine the minimum abscissas of the curve. Derive the condition 
which makes the double tangent impossible. 

cos = =F — ; impossible when b > \a. 
4a 

2. Determine, in a similar manner, the value of corresponding to 
the maximum ordinates. 

cos = 5 . 

oa 



The Cardioid. v 

271. This curve is a particular case of the limaeon, in which 

b — 2a, the node at A becoming 
a cusp. 

Putting b = 2a in equation 
(1), Art. 268, we obtain, for the 
polar equation of the cardioid 

r — 2a (cos 6 ±i),„ 

and, from equation (2), 

r = 2a (1 — cos 6), . (1) 

the position of the curve being 
that indicated in Fig. 45. This 




Fig. 45. 



equation may also be written in the form 



g xxxi.] 



THE CARDIOID. 



>95 



r — 4a siir \0, 



(2) 



In like manner, from equation (3), Art. 269, we obtain the 
rectangular equation of the cardioid 



(x- + j' 2 ) 2 — ^ax {x- + /-) — 4a*y 



(3) 



This curve occurs as a particular case of the epicycloid, see 
Art. 296 ; and also of the hypocycloid, see Art. 299. 

Example. 

1. Determine the minimum abscissa, and the maximum ordinate of 
the cardioid. 

Min. abscissa when = 60 ° and r—a\ 
Max. ordinate when = 120 and r = 30. 



The Cartesian Ovals. 

272. If a point move in such a way that fixed multiples of 
its distances from two fixed points have a constant sum or differ- 
ence, the path described will be a 
Cartesian oval. In other words, 
if we denote the distances of a 
moving point P from two fixed 
points A and B by r and r\ and 
the distance ABby e, the locus of 
P will be a Cartesian oval when r 
and r' are connected by the linear 
relation 




Ir ± mr = ± ne, 



CO 



Fig. 46. 



in which /, m, and n denote numerical constants. 

The line AB is an axis of symmetry 7 , since points symmetri- 
cally situated with reference to this line have the same values 
of r and r . 



296 CERTAIN HIGHER PLANE CURVES. [Art. 272. 

Inasmuch as any two of the four equations included in (1) 
may be regarded as differing in the sign of only one term, it is 
evident that the resulting curves cannot intersect, except in the 
particular case when r or r' admits of the value zero ; that is, 
when A or B is a point of the locus. 

Moreover, an infinitely distant point cannot satisfy equation 
(1) if /and m have different values; hence in general the entire 
locus of this equation consists of closed branches or ovals which 
do not intersect. 

273. To derive the polar equation, A being the pole and 
PAB the vectorial angle 6, we have the relation 

r' 2 = r 2 + c* — 2cr cos 6, 

and eliminating r' between this equation and (1), we obtain 

(/ 2 - m*y + 2c(m t cos 6 ± In) r + (n* - m*y = o. . . . (2) 

The two equations included in (2) determine the same 
points; for, when we use the lower sign and put 180 + 6 in 
place of 6, the sign of the coefficient of r is changed, and con- 
sequently the signs of both roots of the equation are changed. 
Hence the double sign in equation '(2) may be omitted, and as- 
suming / and m to have different values it may be written in 
the form 

(3) 





r 2 


— (2a cos 6 + b)r + k = 0, . . . 




in which 








a — 


m*c 


2lnc , 2 n? — n* 




m*-F 





(4) 
274. By transformation of coordinates we obtain 

(V +r - 2ax + ky = p (*> +/), .... (5) 



§ XXXI.] THE CARTESIAN OVALS. 297 



the rectangular equation of the Cartesian referred to A as an 
origin. 

Putting x — o in equation (5) we have 

/ + k = ± by ; 

whence y — ± \b ± -J V(# 2 — 4/'). 

Hence a perpendicular to AB through A [the axis of y in equa- 
tion (5)] cuts the curve in four real points when <£ 2 — 4k is posi- 
tive ; substituting the values of b and k from equations (4) we 
find that b' — 4k is positive when 

/a + ^ _ m * (£) 

is positive, therefore A is within both ovals when this expression 
is positive. Since interchanging r and r' in equation (1) is 
equivalent to interchanging / and m, we infer that a perpen- 
dicular through B cuts the curve when 

m* + n--P (7) 



is positive, and hence that B is within both ovals when this ex- 
pression has a positive value. Now, since the sum of the ex- 
pressions (6) and (7) is 2n\ which is always positive, it is im- 
possible that both these expressions should be negative ; it 
follows that at least one of the points A and B is within both 
ovals, and, consequently, (since they cannot intersect) that one 
oval is entirely within the other as represented in Fig. 46. 

Equations (3) and (5) do not always represent Cartesians as 
defined above, since the values of the ratios of /, m, and n, de- 
termined by substituting in equations (4) given values of a, b, 
and k, may be imaginary. 

275. These curves were first investigated by Descartes. 
Among the properties of the Cartesian ovals, one of the most 
noteworthy is the discovery of M. Chasles ; namely, the exist- 



298 CERTAIN HIGHER PLANE CURVES. [Art. 275. 

ence of a third point C on the axis AB, such that a linear rela- 
tion exists between the distances of a point of the curve from 
any two of the three points A, B, and C, which are called the 
foci. Thus, if we denote the distance CP by r", a relation of 
the form (1) exists between r and r", and a similar relation be- 
tween r' and r". Two of the foci are always within both ovals, 
like A and B in Fig. 46, while the third focus is exterior to both 
ovals. 

Examples. 

1. Show that when / = m the Cartesian becomes an ellipse or an 
hyperbola according as n is greater or less than ?n, A and B being the 
foci. 

tn 

2. Putting I = m and— = e, and c = 2ae, derive from equation (2) 

n 

(Art. 273) the polar equation of the ellipse or hyperbola. 

a(i-e') 



1 — e cos 

3. Show that if n = m the Cartesian becomes the limacon 

r = 2a cos + b, 

and find, in terms of a and b. the value of the ratio — when the limacon 

m 

is regarded as a Cartesian, and also the distance c between the node at 

A and the second focus B. (The third focus in this case coincides 

with A.) l_ __ ± . = _ &_ 

m 2a 4a 

4. Determine from equation (3) the values of Q and r when the 
radius vector becomes a tangent to either oval. 

cos 6 — , and r = \k. 

a 

When these values are real; that Is, when k [see equation (4)] is posi- 
tive, the pole is the exterior focus. 



I XXXI.] EXAMPLES. 299 

5. Show that if equation (3) represents a limacon referred to the fo- 
cus which does not coincide with the node, we must have 

k = (a + -U)\ 

Show also that the curve is the crunodal limacon if, a and b having 
opposite signs, b numerically exceeds 2a ; the acnodal limacon if b is 
between — 2a and zero ; and that the curve is imaginary when b and a 
have the same sign. 

The Cassinian Ovals. 

276. The locus of a point the product of whose distances 
from two fixed points is constant is called the Cassinian Oval, 
from Cassini, the name of the first investigator of this curve. 

Let A and B be the two fixed points, or foci ; let the middle 
point of the line AB be taken as the pole ; denote the distance 
AB by 2a, and the distances of the moving point P from^f and 
B, by p and p' ; then 

p 2 = c? -f r 1 -f 2ar cos (9, 

and p' 2 = a- + r 1 — 2ar cos 6. 

Whence, denoting the constant value FlG 

of the product pp by c\ 

c~~ = (a' + r-) 2 — 4a"r cos 2 9, 
or r 4 -f 2tf 2 (i — 2 cos" 6) r~ + a 4 — c x — o, . . . (1) 

the polar equation of the Cassinian. 

Transforming to rectangular coordinates, we obtain 

(x- 4- /) 2 '+ 2a- (/ - x") + a 4 - c* = O. . . . (2) 

The distance of the point £7, at which the curve cuts the 
axis of y, from A or B, is by the definition of the curve equal 




300 



CERTAIN HIGHER PIANE CURVES. [Art. 276. 



to c. In Fig. 47, c is taken greater than a ; in this case, the 
curve consists of one continuous oval. 

When c < a, the curve does not cut 
the axis of y, but consists of two dis- 
tinct, ovals, as in Fig. 48. 




Fig. 48. 



Examples. 



1. Show that all the real points of the Cassinian are given by the 
equation 

f = - (x" + a") + i/(4*V + c 4 ), 

and thence show that y is imaginary except when the numerical value 
of x falls between the limiting values \/(c? + c 1 ) and V(a* — f)- By 
means of this result, determine when the curve consists of two ovals. 
Show that when c — o the curve reduces to the two points A and B. 

2. From the expression for -— derived from the equation of the 

Cassinian, show that the tangent is parallel to the axis of x where the 
curve cuts the axis of y, and also where it intersects the circle 



x*+/ 



Derive from this result the equations of the double tangents to this 



curve. 



y=±- 



3. By means of the polar equation of the Cassinian, determine the 
values of which make r a tangent to the curve. 

% sm 20 = ± — j • 



§ XXXI.] THE LEMNISCATA OF BERNOULLI. 



301 



The Lemniscata of Bernoulli. 



277. This curve is a particular 
case of the Cassinian, in which 
c — a. The point C in this case 
falls at the origin, and becomes a 
crunode, as shown in Fig. 49. 

Making c — a in equation (1) 
Art. 276, we have 




Fig. 49. 



or 



r 4 + 2cr {1 — 2 cos 2 6) r" = o ; 
r~= 2CI' COS 20 y 



in which a denotes the distance from the centre to either focus. 
The equation is usually written in the form 



a cos 2< 



(0 



a here denoting the semi-axis of the curve ; that is, OD the 
value of r when 6 = o. 

278. From (1), we have 

r" = a 2 (cos 2 6 - sin 2 6), 



or 

whence we have 



r' — a' 



(x- + yy + a*(/-S) = o, ...... (2) 

the rectangular equation of the lemniscata, referred to its cen- 
tre and axes of symmetry. 

If we turn the initial line back through 45 , (1) becomes 



r — a sin 21 



(3) 



302 CERTAIN HIGHER PLANE CURVES. [Art. 278. 

and the corresponding rectangular equation is 

{x* + yj=2a*xy (4) 

When the equation has this form, the coordinate axes are the 
tangents at the node. 



r o v 



Examples. 

1. In the case of the lemniscata referred to its axes, determine by- 
means of equation (1) the point at which the ordinate is a maximum. 

B = 30 ; r — iV2.a. 

2. When the lemniscata is referred to the tangents at the node, 
determine by means of equation (3) the point at which y is a maximum. 

6 = 6o° ; r = ii/12 .a. 

The Spiral of Archimedes, 

X^T^T ^\ 279. If the radius vector of a curve in- 

— I o 1 — crease uniformly, while the vectorial angle 

I / also increases uniformly ; that is, if 

^-^^--'''l^ r= a6, 

the curve generated will be the spiral of 
Archimedes. 

The distance between two whorls, measured on a radius 
vector, is constant and equal to 2na. 

The dotted portion of the curve in Fig. 50 is obtained by 
giving negative values to 6. 

The Hyperbolic or Reciprocal Spiral 
280. This curve is denned by the polar equation 



Fig. 50. 



§ XXXI. i 



THE HYPERBOLIC SPIRAL. 



303 



When = o, r is infinite ; hence there is a point 
at infinity in the direction of the initial line. To / 
ascertain whether the curve has an asymptote, ^7 
we have 



Fig. 51. 



y — r sin 6 — 



a sin 6 



evaluating this fraction for 6=0, we have y — a ; hence the line 

y = a 

is an asymptote. As 6 increases, r decreases, the curve con- 
tinually approaching but never reaching the pole. When is 
negative, r is negative and determines a similar curve, the com- 
plete curve being symmetrically situated with respect to a per- 
pendicular to the initial line. 

The Lituus. 



281. This curve is 
defined by the polar 
equation 

r 2 6 = a\ 



When 6 is zero, r is 
infinite. To find the 
asymptote, we have 

y = r sin 



Evaluating, we find that y — o when 
therefore, itself the asymptote. 




= o ; the initial line is, 




304 CERTAIN HIGHER PLANE CURVES. [Art. 282. 



The Logarithmic or Equiangular Spiral 

282. This spiral is defined by the 
polar equation 

r = az n \ . . . . (1) 

or log r = log a + n@, 

the logarithm of the radius vector being 
a linear function of the vectorial angle. The shape of the 
curve is indicated in Fig. 53. 

It is proved in Art. 318 that this curve cuts its radius vector 
at a constant angle, hence it is sometimes called the Equiangu- 
lar Spiral. A curve that cuts a system of lines or curves at a 
constant angle is called a trajectory of the system ; hence this 
spiral is the trajectory of a system of straight lines passing 
through a common point. In (1) n is the cotangent of the con- 
stant angle, which is reckoned from the positive direction of the 
radius vector to the direction of the motion of the generating 
point when 6 is increasing. In Fig. 53 this angle is acute, n 
being positive. 

Example. 

Prove that r = ae n0 and r = 6e n9 represent the same spiral, the 
vectorial angles corresponding to equal values of r having a constant 

difference equal to — lo s 7- . 

. n b 

The Loxodromic Curve and its Projections. 

283. A trajectory of the meridians of any surface of revolu- 
tion is called a Loxodromic curve. 

The track of a ship when the course is uniform is a loxo- 
dromic curve traced upon a sphere, and is often called a Rhumb 
line. 



§ XXXI.] THE LOXODROMIC CURVE. 305 

If we project this curve stereographically upon the plane of 
the equator the meridians will project into straight lines, and, 
since in this projection angles are unchanged in magnitude, the 
projection of the curve will make a constant angle with the 
projections of the meridians and will therefore be an equiangu- 
lar spiral. 

Let 6 denote the longitude of the generating point mea- 
sured from the point at which the curve cuts the equator, and 
C the course ; that is, the constant acute angle at which the 
curve cuts the meridians, the generating point being supposed 
to approach the pole as 6 increases. Taking as the pole the 
projection of the pole of the sphere, the polar equation of the 
projected curve will be of the form 

r — az n \ (1) 

in which a is the radius of the sphere, since 6 = gives r = a ; 
we also have 

n = — cot C, (2) 

since the angle whose cotangent is n is the supplement of C 
(see the preceding article). 

Denoting by (f the co-latitude of the projected point we 
have, by the mode of projection, 

- = tan tf ; (3) 

a 

and, denoting the corresponding latitude by /, 

y = {"-¥■ 

Equation (i) is therefore equivalent to 

tan(i*-il) = e-' c ° tc ; 

whence, solving for 6, we have 



306 CERTAIN HIGHER PLANE CURVES. [Art. 283. 

6 — — tan C logetan {\rc — J/) = tan £71og € tan {\n + J/), 
or, employing common logarithms and expressing 6 in degrees, 
0° = 131.9284 tan C- log IO tan (45 ° + J/). . . (4) 

284. 77^ equation of the orthographic projection of the loxo- 
dromic curve on the plane of the equator may be obtained in the 
following manner. 

Denoting the radius vector of this projection of the curve 
by p, we have 

p = a sin (f> = a cos /. (5) 

To obtain a relation between p and 6 we first derive a relation 
between r and p; thus, from equation (3) we have 

r a 

- — tan i$, and - = cot \<i> ; 

a r 

u r a 1 2 

hence - + - = • -, / n ~ - a > 

a r sinJ0cos^0 sin <p 

therefore, by equation (5), 

r a _ 2a # 
a r p ' 

and, eliminating r by means of equation (1), we have 

2a= p(e" 9 + £- ne ) . (6) 

Passing to rectangular coordinates, we obtain 

/ y y\* 

2a = V(x 2 + /) [f**-^ + tr*»*- x 7) m ... (7) 



* This curve is one of Cotes Spirals. For a discussion of these spirals, see Dy- 
namics of a Particle, by Tait and Steele, pp. 147-150, Fourth Edition, London, 

1878. 



§ XXXI.] 



THE PARABOLIC SPIRAL. 



307 



The Parabolic Spiral. 
285. If the axis of the parabola 

y = 4cx . . . 



(0 



be conceived to be wrapped round a circle whose radius is a, 
and the ordinates corresponding to each point be laid off in 
the direction of the radius of the circle, the curve thus de- 
termined will be the. parabolic spiral. 

Taking the pole at the cen- 
tre of the circle, and the radius 
passing through the vertex of 
the parabola as the initial line, 
we have 

x — a6 and y = r — a. 



Substituting these values of x 
and j' in equation (1), we obtain 
the polar equation 




Fig. 54, 



(r — df = 4ca0. 



(2) 



The curve consists of two branches ; the one determined by 
the positive values of r — a is an infinite spiral without the 

circle ; the other branch passes through the centre when 6 = • — , 

4C 

and emerges from the circle at the point at which 



and 6 = 



3 o8 



CERTAIN HIGHER PLANE CURVES. [Art. 286. 



The Logarithmic or Exponential Curve, 
286. The curve defined by the equation 



y 



= e J 



■x 



is called the exponential curve; and, since the 
equation can be written in the form 

Fig. 55. x = log y, 

it is also sometimes called the logarithmic curve. The axis of 
x is an asymptote since £~°°= o. 
The curve defined by 

y = a x , or x = log a y, 

is of the same general form. It passes, through the point (o, i) 
with an inclination whose tangent is log a. 



The Sinusoid. 




287. This curve is defined by 
the equation 



y — b sin - . 
a 

It consists of an infinite number 
of portions similar to the one 
shown in Fig. 56. 
When a — b — 1, we have 

y = sin x. 

The curve corresponding to this particular case may be dis- 
tinguished as the curve of sines. 



§ XXXI. i 



7W.fi CYCLOID. 



309 



The Cycloid. 



288. The path de- 
scribed by a point in 
the circumference of a 
circle which rolls upon 
a straight line is called 
a cycloid. The curve 
consists of an unlimited 
number of branches cor- 




Fig. 57. 



responding to successive revolutions of the generating circle ; a 
single branch is, however, usually termed a cycloid. 

Let 0, the point where the curve meets the straight line, 
be taken as the origin, let P be the generating point of the 
curve, and denote the angle PCR by ip. Since PR is equal to 
the line OR over which it has rolled, 

0R = PR = ai/:, 

and, from Fig. 57, we readily derive 



x = a (ip — sin ip) 
y = a (1 — cos ip) 



(I) 



289. These two equations express the values of x and y in 
terms of the auxiliary variable ip, and constitute the equations 
of the cycloid. If desirable, is easily eliminated from equa- 
tions (1) and an equation between x and y obtained. Thus, 
from the second equation, we have 



cos^ = 



y 



whence sin i/> 



V(2 ay — y 9 ) 
a 



and hence from the first of equations (1) 



3IO CERTAIN HIGHER PLANE CURVES. [Art. 289. 

^nztfcos- 1 — V(2ay — j/ 2 ), .... (2) 

or x — a vers -1 - — V{2ay — y 2 ). 

Equations (1) will in general be found more convenient than 
equation (2). Thus we easily derive from (1) 

dy _ sin ip dip sin tf> 

dx~~ (1 — cos*/;) dip ~~ 1 — cosip' 
whence 

*/ 2 j/ ^ /</k \ _ cos ip — 1 dip _ 1 



<^f 2 ^t: WW (i — cos tpy dx a (1 — cos ^) 2 " 

290. The cycloid is frequently referred to the middle point 
O or vertex of the curve as an origin, the directions of the 
axes being turned through 90 . 

Denoting the coordinates referred to the axes OX' and 
O'Y'j in Fig. 57, by x and y', we have 

y — x — an = a (ip — n — sin ip) } 
x' — 2a — y — a (1 + cos tp), 

or, denoting tp — n by ^', 

y. = a(t/>' + sin #') ) , 

■ y-^(i - cos^') y y J 

In these equations ip' = o gives the coordinates of the ver- 
tex and tp' = ± n gives those of the cusps. 

Example. 

1. Prove that the chord RP joining the point of contact and the 

generating point is perpendicular to the tangent to the cycloid at P. 



§ XXXI." : THE COMPANION TO THE CYCLOID. 3 11 



The Companion to the Cycloid. 

291. This name was given by Roberval, one of the early in- 
vestigators of the cycloid, to the curve described by the point 
M in Fig. 57. 

The equations of the curve are, obviously, 

x = aip, 

and y = a (1 — cos ip) ; 

whence, eliminating ip, we have 

/ 
y = a I — cos- 
V a 

for the rectangular equation of this curve. 

Example. 

1. Show that the companion to the cycloid is a sinusoid symmetri- 
cal to the line y = a, and that it bisects the area of the rectangle 
00'. See Fig. 57. 

The area between the two curves regarded as generated by the variable 
line PM parallel to OX is readily perceived to be equal to the area of the 
semicircle, which would be generated by this line were the point M to de- 
scribe a line perpetidicular to OX. In this way Roberval proved that the 
area of the cycloid is three times that of the generating circle. 

The Ti'ochoid. 

292. The trochoid is the general term applied to the curve 
described by any point in the radius of a circle rolling on a 



312 



CERTAIN HIGHER PLANE CURVES. [Art. 292. 



Fig. 58. 



straight line. If the generating point is taken within the roll- 
ing circle, the trochoid described is called the prolate cycloid ; 
if without the circle, it is called the curtate cycloid. 

Denoting by b the 
distance of the gene- 
rating point from the 
centre of the rolling 
circle, and employing 
the notation given in 
X Fig. 58, the equations 




Prolate cycloid. 



of the curve are 



x — aip — b sin ip 
y — a — b cos if) 



(1) 



When b < a, the curve 
is the prolate cycloid ; 
when b = a, the cycloid ; 
and, when b > a, the cur- 
tate cycloid. 



Fig. 59. 




Curtate cycloid. 



Examples. 



1. Show that the curtate cycloid cuts the axis of x at right angles, 
and, in general, that the line RP is perpendicular to the tangent to 
the trochoid at P. 

2. Determine the points of inflexion in the prolate cycloid. Show 
that at the point of inflexion the radius of the generating circle is tan- 
gent to the curve. . , b 

ib = cos -1 — . 

d\ . a 

To find —4 in terms of tj\ use the method exemplified in Art. 289. 



§ XXXL] 



THE EPICYCLOID. 



313 



The Epicycloid. 



293. When a circle, tan- 
gent to a fixed circle exter- 
nally, rolls upon it, the path 
described by a point in the 
circumference of the rolling 
circle is called an epicycloid. 

Taking the origin at the 
centre of the fixed circle, 
and the axis of x passing 
through A, (one of the posi- 
tions of P when in contact 
with the fixed circle,) a, b, tp, 
and Xj being defined by the 
diagram, we have, evidently, 




Fig. 60. 



*f = &X-:x = -£ f- 



The inclination of CP to the axis of x is equal to tp + X> or * 

- b 

; — ip ; the coordinates of P are found by subtracting the pro- 



jections of CP on the axes from the corresponding projections 
of OC; hence 



/ 7\ 1 7 a + b t 

x — (a + 0) cos ip — b cos — - — tp 



y — (a 4- b) sin tp — b sin 



b 

a 4- b 



r' 



(1) 



These are the equations of an epicycloid referred to an axis 
passing through one of the cusps. 



3H CERTAIN HIGHER PLANE CURVES. [Art. 293. 

Were the generating point taken at the opposite extremity 
of a diameter passing through P in the figure, the projection of 
CP would be added to that of OC; the axis of x would in this 
case pass through one of the vertices of the curve, and the 
second terms in the above values of x and y would have the 
positive sign. 



Algebraic Forms of the Equations. 

294. When a and b are incommensurable, the number of 
branches similar to that drawn in Fig. 60 is unlimited, and the 
curve is transcendental like the cycloid, since the number of 
points in which it may be cut by a straight line is unlimited. 
If, on the other hand, a and b are commensurable, another 
cusp will fall upon A after one or more circuits of the rolling 
circle, and the curve will begin to repeat itself. In this case, 
the curve will be algebraic, for the elimination of ip will give an 
algebraic relation between x and y. 

For example, when b = \a the equations are 



x = \a cos^ — \a cos 3^ 
y — *a sin ip — \a sin yp 



\ 

Employing the trigonometrical formulas 



(1) 



cos yp = 4 cos 3 ^ — 3 cos^> 
sin $ip = 3 sin^ — 4 sin 3 ^, 

we obtain 

x = 3a cos tp — 2a cos 3 (p = a cos ip [3 — 2 cos 2 ^] 



y = 2a sin* ip, whence sin^ 



2a 



§ X X X I .] AL GEBRA IC FORMS OF THE EQl 'A TIONS. 3 I 5 

and eliminating //• 

'-<-(S r I[—(£)7-'+'4r^ 

or 4 (x 2 +/■ - a-) 3 = 277V (2) 

Since this curve has two cusps, it is called the two-cusped epicy- 
cloid. 

The remarks at the beginning of this article apply likewise 
to the curves defined in the next two articles. 



The Epitrochoid. 



295. If a circle rolls 
upon a fixed circle, the 
curve described by any 
point in its plane is called 
an epitrochoid. 

Let c denote the dis- 
tance of the generating 
point from the centre of 
the rolling circle ; and let 
?p be defined as in Art. 293. 
To find the coordinates, 
we subtract the projec- 
tions of c from the corre- 
sponding projections of OC. Whence 




Fig. 61. 



x — (a + b) cos ip — c cos 
y — (a -I- b) sin ip — c sin 



a + b 
a + b 



(1) 



316 



CERTAIN HIGHER PLANE CURVES. [Art. 295. 



If c be greater than b the curve will contain loops, and will 
differ from that drawn in Fig. 61 as the curtate differs from the 
prolate cycloid. 

296. When the fixed and the rolling circles are equal the 
epitrochoid becomes a limacon, and the epicycloid becomes a 
cardioid. For, taking as the pole a point at 
the distance c on the right of the centre, it 
is obvious from Fig. 62 that 6 = ip, and that 

r = 2a — 2c cos 6, 



which is the polar equation of the limacon. 
See Eq. 2, Art. 268. When c = a we have 
the equation of the cardioid. See Art. 271. 




Fig. 62. 



Examples. 

1. Determine the points of inflexion in the epitrochoid. 

ac 2 4- be" + b 3 



cos x 



be (a + 2b) 



2. The point of inflexion is impossible when the value of cos j de- 
termined in Ex. 1 exceeds unity ; find the limiting values between 
which c must fall in order that the points of inflexion may be possible. 

72 

The limits are b and — ■ — ;. 
a + b 

3. Prove that, if c = a + b, the polar equation of the epitrochoid, 
O Y' being the initial line, is 

r = 2 (a + b) sin — ■ — z & . 
v ' a + 2b 



g XXX I.] THE HYPOCYCLOJD AXD HYPOTROCHOID. 317 



The Hypocycloid and the Hypotrochoid. 



297. When the rolling 
circle has internal contact 
with the fixed circle, the 
curve generated by a point 
on the circumference is called 
the hypocycloid, whether the 
radius of the rolling circle be 
greater or less than that of 
the fixed circle. Curves gen- 
erated by points on the ra- 
dius, either within or without 
the circumference of the roll- Fig. 63. 

ing circle, are called JiypotrocJioids. 

Adopting the notation used in deducing the equation of the 
epitrochoid, we have (see Fig. 65 1 



Y 






/ ' \ ^ s ^ 
/ v \ ^^ 

\ 




A 


' V 




X 



OC — a — b, and x = jt 



The inclination of CP to the negative direction of the axis of 
x is 



a — b , 



hence the equations of the hypotrochoid are 



/ / a — b , ^\ 

x ~ {a —o) cos ip -f c cos — 7 — \p \ 

l ■ , . a — b , 

y= a —o) sin '/• — c sin — — — >/• I 



. • . (0 



318 CERTAIN HIGHER PLANE CURVES. [Art. 297. 



When c = b, we have the equations of the hypocycloid. 

/ i\ , 7 a — b , 

x = {a — b) cos tp + b cos — 7 — ip 

y — (a — b) sin ip — b sin ■ — 7 — ip 



. (2) 



If the curve be referred to an axis passing through a vertex, 
the signs of the second terms in the values of x and y given 
above will be reversed as in the case of the epicycloid. See 
Art. 293. 

298. When b = \a, equations (1) become 

x = (b + c) cos tp and y — (b — c) sin tp ; 

whence, eliminating tp, we have 

x* y 



(^ + ^) J (^ - c) 

the equation of an ellipse. Hence when the diameter of the 
rolling circle is half that of the fixed circle the hypotrochoid 
becomes an ellipse. If we put c = b, the above value of y be- 
comes zero ; hence in this case P (Fig. 63) describes the axis of 
x ; that is, the diameter of the fixed circle. 

It follows that all the points of the circumference of the roll- 
ing circle describe diameters of the fixed circle, we may there- 
fore dispense with the circles and produce the same motion by 
constraining two points of a moving plane to describe inter- 
secting straight lines on a fixed plane. The motion thus pro- 
duced is known as tram motion ; it follows that in tram motion 
all the points of the moving plane describe either straight lines 
or ellipses. 

299. When the radius of the rolling circle is double that of 
the fixed circle as in Fig. 64 its centre C describes the circum- 
ference of the fixed circle. Let Pbea point on the circumfer- 



§ XXXL] THE HYPOCYCLOID AND HYPOTROCHOID. 3 19 



ence of the rolling circle, and A the point of the fixed circle 

with which it originally coincided ; then, by 

the definition of the curve, the arc BP is 

equal in length to the arc BA, and, since its 

radius is double that of BA, the angle BCP 

is half the angle BOA. But BCA is half 

BOA ; hence the angles BCP and BCA are 

equal, and consequently C, A, and Pare on 

the same straight line as represented in the 

figure. Hence by the definitions given in 

Art. 268 and Art. 271 P describes a lima- 

con and P describes a cardioid. 




Fig. 64. 



Examples. 

1. Show that if b > a in the case of the hypotrochoid, the curve 
may also be generated as an epitrochoid. 

Put — : — ib = — tb r : then ib = ?//. The constants for the 

b T b — a 

curve as an epitrochoid are a* = — , b' = -(b — a), and c' = b — a. 

' 2. Show that when b < a, in the case of the hypotrochoid, the 
curve may be generated as an hypotrochoid with other values of the 
constants. 

Put — : — rj) = ib' ; then ih = -?//. The new constants are 

b T T a — b ' 

a'= — , b' = T (a — b), and c' = a — b. Since -r = , we have 

b b ' a a 

V b 

- + - = 1 ; 

a a 



hence, if, in one of the two modes of generation, the ratio of the radius 
of the rolling circle to that of the fixed circle exceeds one-half, in the 
other it is less than one-half. 



320 



CERTAIN HIGHER PLANE CURVES. [Art. 3OO. 



The Four-Cusped Hypocycloid. 




Fig. 65. 



300. In the case of the 
hypocycloid when b = \a, the 
circumference of the rolling 
circle is one-fourth the circum- 
ference of the fixed circle, and 
the curve will have a cusp at 
each of the four points where 
the coordinate axes cut the 
fixed circle, as represented in 
Fig. 65. 

On substituting \a for b 
equations (2) Art. 297 become 



x=.\a cos tp + \a cos yp 
y — \a sin tp — \a sin yp 



(0 



Substituting the values of cos $ip and sin 3^ from the for- 
mulas, 

cos yp = 4 cos 9 tp — 3 cos ip, 



and 



sin 3^ = 3 sin tp — 4 sin 3 ^, 



we have 



x — a cos 3 tp 
y — a sin 3 tp 



(2) 



whence 
and 



X^ = #3 cos 2 ^, 
y\ = #t sin 2 ^. 



XXXI. 



THE FOUR-CUSPED HYPOCYCLOID. 



321 



Adding, we have 



2 % 



(3) 



the rectangular equation of the curve. This equation, when 
freed from radicals, will be found to be of the sixth degree. 



Example. 

1. In the case of the four-cusped hypocycloid, express in terms of 
?/' the tangent of the inclination of the chord BP, Fig. 65, and prove 
that this chord is perpendicular to the tangent to the curve at the 
point P. 



The Involute of the Circle, 

301. The curve described 
by any point in a straight 
line which rolls upon a curve 
is called an involute of the 
given curve. The curves de- 
scribed by different points 
of the rolling tangent usu- 
ally differ in shape, but 
when the given curve is a 
circle the involutes differ 
only in position. 

Let BP be one position 
of the rolling tangent to a circle, and let the axis of x pass 
through A, the position of the generating point when in con- 
tact with the circle, then 

BP= aib. 




Fig. 66. 



By projecting the broken line OBP on the coordinate axes, we 
have, OBP being a right angle, 



322 CERTAIN HIGHER PLANE CURVES. [Art. 3OI. 

x — a cos ip + aip sin ip ) , , 

y = a simp — aip cos ip ) 

If the tangent roll back beyond the initial position, a cusp will 
be formed at the point A, and the curve will consist of two 
symmetrically situated infinite branches, as in Fig. 66. 



Example. 

1. Show that the tangent to the involute of the circle is perpen- 
dicular to the rolling tangent, and find the maximum ordinate and 
abscissa in the first whorl. 

Max. ordinate when ip = 7t; 

Max. abscissa when tp = f n. 



The Catenary. 
302. The transcendental curve defined by the equation 



y = - 2 (f+e 



is called the catenary, because it is the form assumed by a 
chain, or perfectly flexible cord of uniform weight per linear 
unit, when suspended from two fixed points. 

The curve is evidently symmetrical with reference to the 
axis of y, and cuts it at the distance c above the origin. 

This curve was first noticed by Galileo, but its true nature 
was first discovered by James Bernoulli. 



§ XXXI.] 



THE TRACTRIX. 



^3 



The Tractrtx. 

303. If a heavy body situated at A on a horizontal plane 
be attached to a string of fixed length a, the other end being 
drawn along a straight line OA\ 
it will describe a curve having 
the characteristic property that 
the intercept on the tangent be- 
tween the point of tangency 
and the axis OX has a constant 
length. 

This curve is called the trac- 
trtx or tractory. The property mentioned above is expressed 
by the differential equation 

^= + y 

dx * V(a'-f) y 
and the equation of the curve is 

a + V(a 2 




± x = a log 



f) 



y 



-V(a--/). 



The curve consists of four symmetrical infinite branches forming 
cusps at A and B, and asymptotic to the axis of x. 

Curves of Pursuit. 

304-. If, while a point Q moves uniformly from O in the 
straight line OQ, P, start- 
ing from A, moves uni- 
formly in the direction 
PQ, the path of Pis called 
a curve of pursuit. 

The characteristic pro- 
perty of these curves is 
that the length of the arc 
AP has a fixed ratio to 
OQ. 




Fig. 63. 



324 CERTAIN HIGHER PLANE CURVES. [Art. 304. 

Denoting the constant ratio -j~ by e, and supposing e not 
equal to unity, the equation is* 

2 (x - ae \ = yXJr€ - ayT ~ e (i\ 

\ 1 — e 2 / a e (i + e) 1 — e ' w 

The axis of x is an asymptote or a tangent, according as e is 
greater or less than unity. 

If e — 1, equation (1) takes an indeterminate form; in this 
case, the equation of the curve is 

^ax + a^—f — 2# 2 log- (2) 



Roulettes. 

305. When any curve rolls upon a fixed curve, every point 
in the plane of the rolling curve describes a curve in the fixed 
plane. Curves generated in this manner are termed roulettes. 

The cycloid, the trochoids, and the epitrochoids are ex- 
amples of roulettes in which the rolling curve is a circle. In- 
volutes, as defined in Art. 301, are roulettes in which a straight 
line takes the place of the rolling curve. 

Inverse Curves. 

306. If a fixed point O in the plane of a given curve be 
taken as the pole, and on the radius vector OP a point P be so 
taken that OP- OP is constant, the curve described by P is 
called the inverse of the given curve with reference to the 
point O. 

* See Dynamics of a Particle, by Tait and Steele, Art. 32-36. Fourth edition, 
London, 1878. 



§ XXXI.] INVERSE CURVES, 3 2 5 

The point is called the centre of inversion, and the value 
of the constant product is called the modulus. It is evident 
that the curve described by P is also the inverse of that de- 
scribed by P. 

The equation of the inverse of any curve is readily derived 
from the polar equation of the curve referred to the centre of 
inversion. Thus, the polar equation of the cardioid referred to 
its cusp is (Art. 271) 

r —2a{\ — cos 6) ; 

whence taking 40? as the modulus we have, for the polar equa- 
tion of the inverse curve, 

2a 



1 - cos 6 ' 

the equation of a parabola referred to its focus. 

Again the polar equation of the cissoid is (Art. 255) 

sin 2 # 
r = 2a 



cos 9 ' 
Hence, taking the same modulus 4a 2 , we have 



cos 6 
r — 2a — 



sin 2 # ' 
or in rectangular coordinates 

y = 2ax, 

the equation of a parabola referred to its vertex. 

It follows therefore that the inverse of the parabola is a 
cardioid when the focus is the centre of inversion, and a cissoid 
when the vertex is the centre of inversion. 



326 CERTAIN HIGHER PLANE CURVES. [Art. 306. 



Examples. 

1. Prove that, the modulus being a 2 , the strophoid is its own in- 
verse, when the vertex A, Fig. 42, is the centre of inversion ; but that 
when its node is the centre of inversion its inverse is an equilateral 
hyperbola. 

2. Prove that, when the node is taken as the centre of inversion, 
the inverse of the limacon is a conic of which this point is a focus. 

3. Prove that the Cartesian is its own inverse with respect to either 
focus. 

4. Show that, when the Cassinian consists of two ovals as in Fig. 
48, each oval may be regarded as its own inverse with reference to the 
centre O. 

5. Show by means of equation (1) Art. 277, or by equation (3) 
Art. 278, that the inverse of the Lemniscata is the equilateral hyper- 
bola having the same axis. 

6. Prove that the inverse of the spiral of Archimedes is the hyper- 
bolic spiral ; and that that of the equiangular spiral is the same curve 
in a new position. 

Pedals. 

307. The locus of the foot of a perpendicular from a fixed 
point upon the tangent to a given curve is called the pedal of 
the given curve with reference to the fixed point. This point 
is called the pedal origin. 

The pedals of a given curve are identical inform with the rou- 
lettes described when a curve equal to the given curve rolls upon it, 
the points of contact being corresponding points of the two curves. 
For the fixed and rolling curves are symmetrically situated with 



> ; XXXI.] PEDALS. 327 

reference to the common tangent ; hence the straight line join- 
ing the point that describes the roulette with the correspond- 
ing point in the plane of the fixed curve is perpendicular to the 
common tangent, and double the length of the perpendicular 
whose extremity describes the pedal. The pedal is therefore 
similar to the roulette. 

308. The pedal of a pedal with reference to the same origin 
is called the second pedal of the given curve ; in like manner the 
third and higher pedals are formed. The curve of which a 
given curve is the pedal is called its negative pedal. 

One of the methods of deriving the equation of a pedal will 
be found in Art. 327. See also Art. 350. 

The method of deriving the equations of negative pedals 
will be found in Art. 381. 

Reciprocal Citrvcs. 

309. If upon OR, the perpendicular from a fixed point O 
upon a straight line, a point P' be so taken that 

OR ■ OP' = k\ 

(k denoting a constant), the point P' is called the pole of the 
straight line with reference to the point O. The given line is 
called the polar of P . 

If now the straight line is the tangent at P to a given curve 
the locus of R is the pedal of this curve ; that of P' is therefore 
the inverse of the pedal. It will be shown hereafter (Art. 384) 
that the locus of P bears the same relation to the locus of P' ; 
that is to say, each of these curves is the locus of the pole of 
the tangent to the other curve ; hence these curves are called 
reciprocal polars, or simply reciprocals* 

* The pole of a straight line and the polar of a curve are frequently defined with 
reference to any conic. In the system above described, which is the one chiefly in 
use, the conic of reference becomes a circle whose centre is and whose radius is k. 



3^8 CERTAIN HIGHER PLANE CURVES. [Art. 309. 

Methods of deriving the equation of the reciprocal will be 
found in Art. 385. 



The Inverse and Pedal of the Conic, 

310. It will be shown in Art. 386 that the reciprocal of a 
conic is another conic, hence it follows from the preceding 
article that the pedal of a conic is the inverse of another 
conic. The inverse and the pedal of the conic therefore consti- 
tute the same class of curves. These curves always have a node 
at the centre of inversion or pedal origin. They are generally 
of the fourth degree and constitute a subdivision of an impor- 
tant class of curves known as Bicircular Qtiartics. When the 
centre of inversion is on the curve, they are of the third degree 
and belong to the class called Circular Cubics. 

The application of the Differential Calculus to Evolutes, 
Envelopes, and Caustics, will be found in Chapter X. 



CHAPTER X. 

Applications of the Differential Calculus to 
Plane Curves 



XXXII. 
The Equation of the Tangent. 

311. THE equation of a curve being given in the form 
y=f(x\ the inclination of the tangent at any point is deter- 
mined by the equation 

, dy - tt x 

tan o — -j- —f{x). 
ax 

Hence, if (*i,_y,) be a point of the curve, the equation of a 
tangent at | x . y-_< will be found by giving to the direction- 
ratio in, in the general equation 

y — )\ - m (x — x x ), 

the value -f- > ; thus 
dxJ x% 

y - y -a%jf~ X ^ (I ) 

0^ y-y,=f [x t ) {X - X t ) (2) 

For example, in the case of the semi-cubical parabola 

y l — ax\ 



130 APPLICATIONS TO PLANE CURVES. [Art. 3 1 1. 



we have -~- — § A/ — 



dx 6 y x 

The point (a, a) is a point of this curve ; the equation of 
the tangent at this point is, therefore, 

y - a = f (x - a), 

or 3j — 2x = a. 

If the given equation is not in the form y =f(x), we can 

still use equation (1) ; but, since -j- will then be expressed in 

terms of x and y, the value of the derivative must evidently 
be obtained by substituting in this expression the given simul- 
taneous values of x x and y x . 

To illustrate, let the equation of the curve be 

xy h —?>xy + 6y* + 2x — o, 

and let it be required to find the tangent at the point (2, 1), 
which is a point of the curve since these values of x and y 
satisfy the given equation. 
Differentiating, we obtain 



whence 



dy 


y z — 6xy + 2 


dx 


3 (xf - x* + 4y) ' 




±1 =f 

dxA (2, 1) 



The equation of the tangent at this point is, therefore, 
j, - I = f(.r- - 2). 



§ XXXI I.] THE EQUATION OE THE X OK MAI.. 33 1 



The Equation of the Normal. 

312. A perpendicular to the tangent at its point of contact 
is called a normal to the curve. 

The coordinate axes being rectangular, the direction-ratio 
of the normal is the negative reciprocal of that of the tangent ; 
for the inclination of the normal is \n + (ft, and 

tan (|7T + </)) — — cot <j>. 

The equation of the normal may, therefore, be written thus — 

dx 



y-yx = --7- 

dy 



(*-*i), (i) 



As an illustration, let us take the equation of the ellipse 

. dy b 2 x 

whence -~-= — ■ . 

dx ay 

The equation of the normal at any point (x lt y x ) of the ellipse 
is, therefore, 

a *y^ ( \ 



The Equation of the Tangent to a Conic. 

313- The general equation of the second degree may be 
written in the form 



332 APPLICATIONS TO PLANE CURVES. [Art. 3 1 3. 

Ax* + 2Bxy + Cf + 2Dx + 2Ey + F = o; 

from this we derive 

dy _ _ ^;tr + 2?J/ + D 
~dx ~ Bx + Cy + E ' 

The general equation of a line tangent to the curve at the 
point (x 1} y,) is, therefore, 

(y-y x )(Bxx + Cy x + E) + (x - x 1 )(Ax 1 + By, + D) = o, (1) 

which reduces to 

^(^ + Cy x + E) + x(Ax, + By, + Z>) 

- (Ax* + 2^/, + 6>! 2 + Ztr, + £>,) = o ; (2) 

but, since (x lf y,) is a point of the curve, we have 

Ax, 2 + 2Bxy x + 6> x 2 + 2Dx x 4- 2^ 4-^=0,. . . (3) 

and, by adding (2) and (3), we have 

x (Ax, + By, + D) + j/(^*ri + Cy x + £) + Z^ + Ey x + F=o, (4) 

the equation of the tangent expressed in terms of the coordi- 
nates of the point of contact. 

This equation may also be written in the form 

Axxx + B(xxy + xy x ) + Cyy, + D (x + x,) + E (y + y,) + F=o. 

Subtangents and Subnormals. 

314. Denoting by s the length of the arc measured from some 

ds 
fixed point, — denotes the velocity of P, the generating point 
dt 



I XXXIL] SUBTANCENTS AXD SUBNORMALS. 



333 



of the curve : let PT. equal to ds y be measured on the tangent 
at P t then PO and OT "will represent dx and dy. and the angle 
77Y? will be ;' : hence t 



and 



, dx ■ ± dy , \ 

coso=— . sino=f, (i) 



^ = | ... (2) 




A 



dx 



Fig. 69. 



315. 



T R 



The distance PT Fig. 70) on the tangent line inter- 
cepted between the point of contact 
and the axis of x is sometimes called 
the tangent, and in like manner the in- 
tercept PX is called the normal. 

From the triangles PTR and XPR, 
we have 




Etc -: 



PT — y cosec 6 = y 



ds_ 
dv 



J\ x Jdx\^ 



~ -1' 



PX = y sec 6 = y — =j j 1 - 



m 



The projections of these lines on the axis of x. that is TR 
and RX. are called the subtangent and the subnormal. 
From the same triangles, we have 



the subtangent. 



~ D , dx 

TR=y cot o= : - . 



and the subnormal, RX= j- tan <5 = j 



;?y 



334 APPLICATIONS TO PLANE CURVES. [Art. 316. 



The Perpendicular from the Origin upon the Tangent. 

316. If a perpendicular/ to the tangent PR be drawn from 
the origin, we have, from the triangles in Fig. 

Y p — jrsin <f> — y cos <f>, . . . (1) 



Jjs/ >f x $ — 90 being taken as the positive direction of p. 



'E 



Substituting the values of sin <f> and cos <f>, 
Fig. 71. equation (1) becomes 

_ xdy — ydx __ xdy — ydx , . 

P '' ~~^~ds _ V(dx*+df) ^ 

For example, let us determine / in the case of the four- 
cusped hypocycloid, 

x — a cos 3 ip, y— a sin 3 ip. 

Differentiating, 

dx — — 3*2 cos 2 ^ simp dip, and dy = $a sin 2 ip cos ip dip ; 
whence ds = 3^ sin ip. cos ip dtp. 

Substituting in equation (2) we obtain 
p — a cos 3 ip sin ip + a sin 3 ip cos ip = a simp cos ip = ?/{axy). 

To ascertain the direction of p it is necessary to determine 
(f). The ambiguity in the value of <f> as determined from the 

equation tan <f> = -~- may be removed by means of one of the 

formulas of Art. 314. Thus, in the present case, we have 

tan (f) = — tan tp, whence (f> = — ip, or <f> = n — ip; 



g XXXII.] THE PERPEXDICULAR IfE TAXGEXT. 

nee co.- = — cos '/•, 

we must take 6 = - — 

The direction of/ when positive is therefore - - . 

Examples XXXII. 
i. In the case of the parabola of the «th degree 
a m ~- = 
find the equations of the tangent and the normal at the point (.:. 

2. Find the subtangent and the subnormal of the parabola 

. ' = - 

3. Prove that the normal to the catenary (Art. 322 ) :s — . 

4. Prove that the subtangent of the exponential curve 

:■ = 

is constant, and find the ordinate of the point of contact when the 
tangent passes through the origin. £. 

5. Find the subnormal of the elli:: e equation is 

6. Find the subtangent of the curve 

a*' 1 = 
also the subtangent of the curve 



— , and nx. 
n 



33 6 APPLICATIONS TO PLANE CURVES. [Ex. XXXI L 

7. In the case of the parabola 

find/ in terms of x. 



y* = 4ax, 



For the upper branch, p = 



V(a + x) ' 

8. Find, in terms of fi, the equation of the tangent to the four- 
cusped hypocycloid (Art. 300), and thence show that the part inter- 
cepted between the axes is of constant length. 

9. Show that all the curves represented by the equation 

,-)■+ (i 

(different values being assigned to n), have a common tangent at the 
point (a, b) ; find the equation of this tangent. 

10. Show that the equation of the tangent to the curve 

at the point (x h j^i), is 

cftx^y + $y$x = cfiftxfy^ • 
and, denoting the intercepts on the axes by x Q and y 0} prove that 

*L 4. yl - T 

a' "*" b* ~ *' 

11. Find the equation of the tangent at any point of the curve 

x n + y n = a n . 

yyn-i + xXi n ~ x = d n . 

12. Find the equation of the tangent at any point of the curve 

x m y f 

nx x y + my x x — (m + n)x x y x . 



§ XXXII.] examples. 337 

13. Find the general equation of a tangent to the conic 



XT — 2V — 4AT — X = O. 

Show that the point (1, — 2) is on the curve, and find the equation of 
the tangent line at this point. 

14. Show that, when the conic passes through the origin, the equa- 
tion of a tangent line at this point, as derived from the general equa- 
tion, Art. 313, is identical with that obtained by the method of 
Art. 218. 

15. In the case of the epicycloid, find the value of ds in terms of 
the auxiliary angle >/\ See Art. 293. 

7 / T\ • ^ J, 

ds = 2 (a + 0) sin — — dip. 

20 

16. Determine the value of p in the case of the epicycloid em- 
ploying the value of ds given in the preceding example. 

/ a • ad) 

p = (a 4- 20) sin — . 



XXXIII. 

Polar Coordinates. 

317. When the equation of a curve is given in polar co- 
ordinates the vectorial angle 6 is usually taken as the inde- 
pendent variable ; hence, denoting by s an arc of the curve, it 
is usual to assume that ds and dd have the same sign ; that is, 

that — is positive. 
du 

In Fig. 72 let PT, a portion of the tangent line, represent 

ds ; then, producing r, let the rectangle PTbe completed, and 



338 



APPLICATIONS TO PLANE CURVES. [Art. 3 1 7. 



let t/y denote the angle TPS; that is, the angle between the 
positive directions of r and s. The re- 
solved velocities of P along and perpen- 

dr 




Fig. 72. 



dicular to the radius vector are — and 



rdd 



dt 



-j--, the latter being the velocity which P 

would have if r were constant ; that is, if 
P moved in a circle described with r as a 
radius. Hence we have 



PS=dr 



and 



PR = rdd. 



From the triangle PST, we derive 



tan i/> = 



rdd 
dr' 



simp = 



rdd 
ds' f 



. dr , . 



and 



7^ 



= ,/[, 



dr\ r 
de) . 



(2) 



318. The second of equations (1) shows that, in accordance 
with the assumption that ds has the sign of dd, the value of i/> 
will always be either in the first or in the second quadrant. 

The first of equations (1) is equivalent to 



cot^ 



dr 

rdd' 



(3) 



which shows that cot ?/; is the logarithmic derivative of r re- 
garded as a function of 6. Thus in the case of the logarithmic 
spiral 



r = as ne 



we have 
hence 



log r = log a + nQ, 
cot ib — n. 



§ XXXIIL] 



POLAR COORDINATES. 



339 



whence it follows that, in the case of this curve, ip is constant. 
See Art. 282. 

319. It is frequently convenient to employ in place of the 
radius vector its reciprocal, which is usually denoted by u ; 
then 

r — -, and dr — (4) 



Making these substitutions in equations (2) and (3) we have 



=y 



ds __ 1 

Te~ u 



duX 



(5) 



and 



cot f = — 



udO 



(6) 



Polar Sub tangents and Subnormals. 



320. Let a straight line perpendicular to 
the radius vector be drawn through the pole, 
and let the tangent and the normal meet 
this line in T and N respectively ; then the 
projections of PT and PN upon this line, 
that is OT and ON, are called respectively 
the polar sub tangent and the polar subnormal. 
In Fig. 73, OPT = ij; ; whence 




OT—r tan if> = r 



dO 
dr 



dd 

du J 



Fig. 73. 



and 



ON =r cot if: = 



dr 



du 



dd u*dd 

Fig. 73 shows that the value of OT is positive when its 



34° APPLICATIONS TO PLANE CURVES. [Art. 320. 

direction is 6 — 90 ; that of ON is, on the other hand, positive 
when its direction is 6 + 90 . 



The Perpendicular from the Pole ttpon the Tangent. 

321. Let/ denote the perpendicular distance from the pole 
to the tangent ; then, from Fig. 73 we obtain 

. . % dS r" , x 



Ar*-<£n ' ' 



ds 
These expressions give positive values for /, because — is 

assumed to be positive, and Fig. 73 shows that/ has the direc- 
tion (f> — 90 , <j) being the angle which the positive direction of 
s makes with the initial line. 

The relation between p and u is obtained thus: — from (1) 
we have 

f ~ r'dO" ' 
and, transforming by the formulas of Art. 319, 

I , (du\ , v 

f = u+ Kw) ; • (2) 

322. The expression deduced below for the function 
u -f -j~- is frequently useful. 
Differentiating (2), we have 

7 du d"*u 2dp 

2uda + 2 d§-He=--f'' 



§ XXXIII.] THE PERPENDICULAR UPON THE TANGENT. 34 1 

hence ( u + -— ) du — 4i 

or since du — — — , 

r 

d'u r" dp 
u 4- — _£_ 

The Perpendicular 2ipon an Asymptote. 

323. When the point of contact P passes to infinity the 
tangent at P becomes an asymptote, and the subtangent 
OT coincides with the perpendicular upon the. asymptote. 
Hence (ft denoting a value of 6 for which r is infinite) the length 

of this perpendicular is given by the expression — 

du. 



and 



like the polar subtangent is, when positive, to be laid off in the 
direction ft — 90°. 

This expression for the perpendicular upon the asymptote 
is also easily derived by evaluating that given in Art. 244. 
Thus — 



rsin (ft - 6) 



sin (ft - ey 



-i*i 



de~ 
du_ 



Points of Inflexion 

324. When, as in Fig. 73, the curve lies between the tan- 
gent and the pole, it is obvious that r and / will increase and 

dp 
decrease together; that is, ~- will be positive. When on the 

other hand the curve lies on the other side of the tangent, 

~- is negative. Hence at a point of inflexion -J- must change 

sign. 



34 2 APPLICATIONS TO PLANE CURVES. [Art. 324. 

Now, since/ is always positive, it follows from the equation 
deduced in Art. 322 that the sign of this expression is the same 
as that of 

d 2 u , N 

u+ d¥' (I) 

hence at a point of inflexion this expression must change sign. 

325. As an illustration, let us determine the point of in- 
flexion of the curve traced in Art. 245 ; viz., 

ad* 
r — 



e*-i 



In this case 



-JO 



, du 2 / _ 3 , d^u 6 „_ 4 

whence — = - 6 ', and — = 6 

dd a dff 1 a 

therefore u + -=£ — - ( 1 _6>~ 2 - 66^) 

dd a\ J 

0* _ 2 - 6 



Putting this expression, equal to zero, the real roots are 

= ± V3, 



and it is evident that, as 6 passes through either of these values, 

, . d'li . 

the expression u + -^ cha 

flexion are determined by 



the expression u + -^ changes sign. Hence the points of in- 



= ± V3 and r = -— 



§ XXXI 1 1.] POINTS OF INFLEXION. 343 



326. When a curve has a branch passing through the origin 
without inflexion, the corresponding value of /, which is zero, 

evidently constitutes a minimum value, and consequently -j- 
7., ' dr 

and u — - change sign. The latter expression in fact passes 

through infinity ; hence the above method is inapplicable at the 
pole. The existence of a point of inflexion situated at the pole 
is however indicated by the fact that two values of r having 
opposite signs become zero for the same value of 6. For ex- 
ample, in the case of the lemniscata 

r = ± a \> '(cos 20), 

both values of r become zero when 6 = 45 and when 6 = 135 ; 
hence this curve has two points of inflexion at the pole. 



The Polar Equation of a Pedal. 

327. Since the locus of R in Fig. 71, or in Fig. 73, is the 
pedal of the given curve with reference to O as the pedal origin 
(see Art. 307),/ is the radius vector of the pedal, and its incli- 
nation to the initial line, which is ^5 — 90 , is the corresponding 
vectorial angle. Hence, if we put 

r T —p and 6 Z = (f> — \n y 

the relation between r T and 6 Z will constitute the equation of 
the pedal. 

Thus in the case of the four-cusped hypocycloid the values 
of/ and (f) determined in Art. 316 are 

/ = a sin ip cos tp, and ^ = n — i/> ; 

whence Q^ — \n — ip . 



344 APPLICATIONS TO PLANE CURVES. [Art. 327. 

Putting r x for/ and eliminating tp, we have 

r x = a cos 6 X sin X = \a sin 20 x . 

328. When the equation of the curve is given in terms of x 
and y, it is convenient to eliminate ^ at once from the value of 
/ given in equation (1) Art. 316. Thus, since <J> — X + }?r, 

r x — x cos 0! + y sin X , (1) 

in which x and j/ are to be expressed in terms of X . For ex- 
ample, if the given curve is the common parabola 

y 1 — 4<zx, 
we have tan $ = — ; 

y 

whence y = 2a cot (f> and x — a cot 3 ^, 

or ^ = — 2a tan # x and x =. a tan 2 ^ . 

Substituting in equation (1) we have 

sin'0 t sin 9 t sin 2 r 

r x = a jr — 2a ^ — — a w , 

cos X cos u x cos u x 

which is the polar equation of a cissoid. Hence the pedal of 
the parabola with reference to its vertex is the cissoid. 

329. When the equation of the given curve is in polar co- 
ordinates, the most convenient arrangement of formulas is that 
given below. By Fig. 73, Art. 320, 

$=z0 + i/>, hence t - 6 + tp - \n. . . (1) 



I XXXIII.] THE POLAR EQUATION OF A PEDAL, 345 

If we eliminate if) from the latter equation by means of the 
formula 

COt * = -rd8' ••••••• « 

we shall have a relation between t and 6. 

Again the value of p (Art. 321) gives 

r x — r sin tb = — ; 5 — — , .... (X) 

7 V(i+cot a ^) u; 

in which cot ip is given by equation (2), and r by the equation 
of the curve. Finally we eliminate 6 from this result by means 
of the relation between 6 and 0,. 

330. For example to find the pedal of the curve 

r m = a m cos md, (1) 

we have log r = log a -\ log cos mO ; 

m 

hence in this case equations (2) and (3) of the preceding article 
become 

cot if> = — tan md, (2) 



, (cos md)'- , '-^ , . 

and r T = a x - — = #(cos md) m . ... (3) 

sec md J J 



From (2) we have ip = md + \n. Now 6 = o in (1) gives 
r~ a and hence determines a real point of the curve, and, since 
sin ip is always positive, we must take ip = \n when 6 = o ; 
therefore 



34-6 APPLICATIONS TO PLANE CURVES. [Art. 330. 

Hence from equation (1) of the preceding article 

6 S = (m + i)d, (4) 

and, eliminating 6 between (3) and (4), 



m 
r x = a cos 



1 \ 171 

+ l) ' 



or r <"' +1 = a '" + lcos {^-^ (5) 



m 

m + i^ 1 



This equation is of the same form as equation (1), and may 
be derived from it by putting in place of m. 

331. The inverse of any curve in this class belongs also to 
the same class of curves. For, from equation (1), we have 



r = a (cos md) m , 

and, taking as the modulus a 2 , we have for the inverse 

1 
r — a (cos md) m ; 

but cosm8 = cos (— md), hence the preceding equation is equi- 
valent to 

r- m = a~ m cos(— md), (6) 

which may be obtained from (1) by writing — m for m. 

Again, since by Art. 309 the reciprocal polar is the inverse 
of the pedal, the reciprocal polar of (1) is the inverse of (5) ; its 
equation is therefore 



m 4- 1 — ^ m + 



w .+ x cos( — — 6 ) • . . . (7) 

\ m + I / v// 



§ XXXIII.] THE POLAR EQUATION OF A PEDAL. 347 

Thus the pedal, the inverse and the reciprocal polar of each 
of the curves of the class represented by equation (i), are like- 
wise members of that class of curves. Many well-known curves 
can be shown to belong to this class. See Examples 21, 22, 
and 23. 

Examples XXXIII. 

1. Prove that, in the case of the lemniscata r 1 = a* cos 2O, 

1 ^ . ds a 2 

f = 20 + } 2 7T, and ^- = ~. 

2. Prove that the polar subtangent of the hyperbolic spiral is con- 
stant. 

3. Find the subtangent and the subnormal of the spiral of Ar- 

W A A +U * ds *V+ r ") 

chimedes, and prove that — = — ■ — . 

dr a 

. a 2 

4. Find the subtangent of the lituus r" = — , and prove that the 

perpendicular from the origin upon the tangent is 

2a VQ 
1(1 +4*V 

5. Find the polar subtangent of the spiral r (e 9 + €~ e ) = a. 



h » _ £ - V 

6. Find the value of/ in the case of the curve r n = a n sin nQ, 

p = a (sin nh) 

7. In the case of the parabola referred to the focus 

2a 



cosS 



prove that/" = ar. 



34 8 APPLICATIONS TO PLANE CURVES. [Ex. XXXIII. 

8. In the case of the equilateral hyperbola 

2 

r 2 cos 20 = a 2 , prove that/ = — • 

9. In the case of the lemniscata 

r s 
r 2 — a* cos 20, prove that/ = — . 



/ 2\ 

10. In the case of the ellipse r — — - — , the pole being at 

the focus, determine/. 



* V(i — 2^cos0 + e*)' 

11. In the case of the cardioid 

r = a (1 + cos e), prove that r 3 = 2ap 2 . 

12. Determine the asymptotes of the hyperbola by the method of 
Art. 323, the polar equation being 

a{e*-i) 

r = — * . 

1 — e cos0 

1 =±sec~ 1 ^, p = + aV(e* — 1). 

13. Prove that the condition which determines points of inflexion 

d*u 

in polar coordinates ; namely, that u + -jj shall change sign, is equi- 

/dr\ 2 d' l r 
valent to the condition that r * ^ 2 \tA ~~ r Ijtf shall change sign. 

14. Determine the points of inflexion of the lituus r% — a*. 

= J, and r = ± a ^2. 



§ XXXI 1 1.] XMPLES. 349 

15. Show that the curve rB m — a has points of inflexion deter- 
mined by = \ [av(i — ;//)]. 

16. Show that the curve rOs'mO = a has a point of inflexion at 
which r = — . 



17. Show that the curve r = bfj n has a point of inflexion deter- 

n 

mined by r = b [— n (n + i)] r . 

18. Show that, if the curve r = 4v~\ nas an asymptote whose in- 
clination to the initial line is If the perpendicular on it will be 

^(80 " 

19. Show that the curve r(20 — 1) = 2^9 has an asymptote de- 
termined by = 1,J> = — £#, and a point of inflexion determined by 
the real root of the equation 20 s — 0~ — 2 = o. 

20. Determine the asymptotes to the curve r = —. r. for 

cos/^ (Q — ay 

the branch which passes through the point (a, a). 

, n a 

This curve is one of Cote's Spirals. See Tait atid Steele's Dynamics, 
fourth edition, London, 1878, p. 150. 

21. Show that if m = — J the general equation of Art. 330 repre- 
sents a parabola referred to its focus, and determine its pedal and its 
reciprocal. 

The pedal is the straight line r cos — a ; 
the reciprocal is the circle r = a cos 0. 

22. Show that if m = % in the general equation of Art. 330 the 
curve is the cardioid whose axis is a, the pole being at the cusp ; and 



35° APPLICATIONS TO PLANE CURVES. [Ex. XXXIII. 

thence determine the equations of the pedal and of the reciprocal to 
this curve. 

t Pedal r — a cos 3 i0; 
reciprocal r = a sec 3 \ 0. 

23. Determine the curves, belonging to the class discussed in Art. 
330,. of which the inverse is identical with the pedal, and consequently 
the reciprocal identical with the original curve. 

m = o, and m = — 2. 

The former gives the circle and the latter the eqidlateral hyperbola each 
referred to the centre. The inverse and pedal of the latter curve is the 
lemniscata. 

24. Show that the ;zth pedal of the curve 

r m = a nt cosz#0 

tn 
is determined by putting for m in the general equation ; and 

mn + 1 

that when n is negative, the same formula gives the successive nega- 
tive pedals. 

25. Find the pedal and the reciprocal polar of the equiangular 
spiral r — ae n0 . 

Each of these curves is a spiral equal to the original spiral, but in a 
different position. 

26. Find the reciprocal polar of the curve r = a sec 3 ^9. 

r T * — a* cos J0x. 

27. Find the pedal of the circle (x — of + y* — d\ 

The limacon r T = b cos 0! ± a. 

28. Find the polar equation of the pedal of the ellipse, the centre 
being the pedal origin. 

n 5 * = a" cos* 0x + ^sin'Gx. 



§ XXXIII.] EXAMPLES. 351 

29. Prove that, in the case of the threc-cusped hypocycloid in which 

a = & 

tan <j) — — tan I fp ; 

and thence find the pedal of the curve. 

r, = — b cos 30i. 

30. Find the pedal of the epicycloid. See Art. 293. 

r x = (a + 2^) sin -^ — 8 . . 



XXXIV. 
Curvature. 



332. If, while a point P moves along a given curve at the 

rate — - , it be regarded as carrying with it the tangent and 

normal lines, each of these lines will rotate about the moving 

point P at the angular rate -— , <j> denoting the inclination of 

the tangent line to the axis of x. 

The point P is always moving in a direction perpendicular 

ds 
to the normal with the velocity — . Let us consider the 

vC motion of a point A on the normal at a given dis- 

tance k from P on the concave side of the arc. 
While this point is carried forward by the motion 

ds 
x of P with the velocity -7 in a direction perpen- 

ig. 74. dicular to the normal, it is at the same time car- 
ried backward, by the rotation of this line about P, with the 




35 2 APPLICATIONS TO PLANE CURVES. [Art. 332. 

velocity —7— ; since this is the velocity with which A would 

aZ 

move if the point P occupied a fixed position in the plane ; 
and the direction of this motion is evidently directly opposite 
to that of P. Hence the actual velocity of A will be 

ds , d<f> 
~di ~ ~di* ' 



in a direction parallel to the tangent at P. 

Let p denote the value of k which reduces this expression 
to zero, and let C (Fig. 74) be the corresponding position of A : 
then, 

ds d</> 

ds 
whence PC ' = p — — - (1) 

dtp J 

333. The value of p determined by this equation is, in 
general, variable ; for, if the point i^move along the curve with 

a given linear velocity — , the angular velocity — - will gene- 
rally be variable. If however we suppose the angular velocity 
~- to become constant, at the instant when P passes a given 

ds 
position on the curve, -yy, the value of p, will likewise become 

constant, and C will remain stationary. When this hypothesis 
is made, the curvature of the path of P becomes constant, for 
P describes a circle whose centre is C, and whose radius is p. 
Hence this circle is called the circle of curvature corresponding 
to the given position of P; C is accordingly called the centre of 
curvature ; and p is called the radius of curvature. 



§ XXXIV.] CURVATURE. 353 



The Direction of the Radius of Curvature. 

334. If in Fig. 74 the arrow indicates the positive direction 
of s ; the case represented is that in which <j> and s increase to- 
gether, and therefore the value of p as determined by equation 
(1), Art. 332, is positive. Hence it is evident that when p is 
positive its direction from P is that of PC in Fig. 74 ; namely, 
(f> + 90 . In other words, to a person looking along the curve 
in the positive direction of ds, p, when positive, is laid off on the 
left-hand side of the curve. 

For example, let the curve be the four-cusped hypocycloid, 

x = a cos 3 ip, y = a sin 3 ip. 

It was shown in Art. 316 that for this curve 

ds — $a sin ip cos ip di/:, and (j) = n — ip ; 

hence d<fr — — dip, 

ds 
and p = —— = — 3# sin ip cos ip (1) 

When is in the first quadrant p is negative ; its direction 
is therefore (j) — \n — \tt — tf>, which is in the first quadrant. 
When ip is in the second quadrant p is positive and its direction 
is (j) 4- \n = I n — 1/:, which is in the second quadrant. 



The Radius of Curvature in Rectangular Coordinates. 

335. To express p in terms of derivatives with reference to 
x, we have 



354 



APPLICATIONS TO PLANE CURVES. [Art. 335. 



= tan" 



1 dy 
dx 



* and 5 = V 



I + 



dy 

Ixl _T 



hence 



dx 



ds 



dy 

dx" 



1 + 



and 



dx 
dx 



dx) 



dy 

dx 1 



(0 



ds 



Since ^- is assumed to be positive, <j) should be so taken as 
dx 

to cause x to increase with s, and it must be remembered that 

the direction of p is <f> + 90 when p is positive, in accordance 

with the remark in the preceding article. 

336. To illustrate the application of the above formula, we 
find the radius of curvature of the ellipse 



y 



Differentiating, 

and 

Putting b = aV(i 



a 

dy __ bx 

dx ~' aViff — x' 2 )' 



dy 

dx 2 



ab 



(a- - x'f 
) we obtain 



(1) 
(2) 
(3) 



1 + 



dy\ l _ c? —e*x 



2 

2~ > 



dx / a'— x' 
whence, substituting in equation (1) of the preceding article, 



§ XXXIV.] THE RADII'S OF ( 7 T R J '. I TL A'/-.'. 355 



* == ^V(i-0 (4) 

337. When this result is applied to a meridian of the earth 
regarded as an ellipse, it is usual to express p in terms of the 
latitude /. 

The definition of /gives 

/ = <p - Iti\ 
hence, from equation (2), 

. , a 2 (a"— x-) 
tan- / = — \—- t — '- 
b'-x- 

rf- 

therefore x 



a 2 


-x* 


(1 - 


•e*)x ' 




a- 



2 /> 



I -f- (1 — r) tan J / sec' / — e 1 tan 2 / 



„ „ <r(sec 2 / — e 1 sec 2 /) a{\ — r) 

and a — ex\ == — ^ — — — — —=-*■ = — . / . 

sec- 1 — c tan- / I — r sin' / 



Whence, substituting in equation (4), 



(l-/sin ! /)f 



(1) 



The Radius of Curvature at the Origin when the Tan- 
gent and Normal are the Coordinate Axes. 

338. When the tangent and normal are the coordinate axes, 
the formula deduced below affords an easy method of deter- 
mining the radius of curvature at the origin. 

If the axis of x is a tangent to the curve at the origin, we 
obviously have 

ds — dx, 



356 



APPLICATIONS TO PLANE CURVES. [Art. 338. 



and the value of p given in equation (1) Art. 335 becomes 

1 



Po = 



dy 

dx' 2 _ 



('•) 



But when the axis of x is a tangent at the origin, we have 

o ; 



dy' 
dx. 



X_A 



therefore *L assumes the indeterminate form - ; hence, 
% _i o 

evaluating, 

dy 

y~\ _dx 

x*A Q 2x 



= i 



dy 



dx" 



(2) 



therefore, by equation (1), 



2jJo 



(3) 



It is to be noticed, when this formula is applied, that <j> is 
zero because ds was assumed equal to dx ; and hence that, by- 
Art. 334, the direction of p when positive is 90 ; in otjier words, 
the centre of curvature is above the origin when p \s positive.* 

Again, when the axis of y is a tangent at the origin, we 
have 



Po 



2JJ 



(4) 



* When the equation of the curve is given in polar coordinates, Jihe initial line 
being tangent at the pole, we have, for the value of p at this point, 



' _ _#n _ r l cos-fl n _ rl 
~' 2y Jo ~ 2^sin9j0=o _ 29_| ( 



XXXIV.] THE VALUE OF p AT THE ORIGIN. 357 



the centre of curvature being on the right of the origin when 
p is positive. 

The method of determining curvature used by Sir Isaac 
Newton in The Principia is equivalent to the employment of 
the above formulas. 

339. As an example let us take the equation of a conic, 

/+ (i — e*)x* — 2a (i — r)x — o. 

This curve is tangent to the axis of j, since the equation of the 
tangent at the origin is x — o. See Art. 218. 
Dividing the equation by 2x, we have 

2- + (i — e*) - — a (i — e, = o, 

2X V J 2 K J 

and putting x = o we derive by equation (4) 

p = a{i - e Q ). 

In applying these formulas when the origin is a node the 
method illustrated below is convenient. Let the equation of 
the curve be 

x'° + ax-jr — ax % y — 2c?xy 2 + ciy z = o, 

y- being a factor of the terms of lowest degree, we employ 
formula (3), whence 

x* 

substituting in the equation of the curve, we obtain 

5 x 6 x b a x b a x G 

x — a — -. — a a' — - -f a~ ——. = o ; 

4P 2 2/3 2f>~ Sp 3 



35^ APPLICATIONS TO PLANE CURVES. [Art. 339. 

dividing by the lowest power of x that appears, and putting 
x = o in the resulting equation, we have 

2p 2 — ap — d = o ; 

whence p = a and p = —\a. 

It follows that the curve has two branches touching the axis 
of x at the origin, one branch being above the axis and the 
other below it. 



Expressions for P in which x is not the Independent 

Variable. 

34-0. To express p in terms of derivatives with reference to 
y, we have 



d'x [ /dx\ r 



u d(f> df L \dy I 

whence —— = — , and p — 



dy 



dy (dxV 7 r d^x 

1 + \dy) df 

In this case ds and dy were assumed to have the same sign, 
hence <f> must be taken so as to causey to increase. 

341. When x and y are expressed in terms of a third vari- 
able we employ the formula deduced below. 
Differentiating 

both dx and dy being regarded as variable, we have 



§ XXXIV.] EXPRESSIONS FOR p. 359 

dx d*y — dy d*x 

, . _ dx* dx d'y — dy d % x u 

1 { /dy\* " dx" I df 

. ds (dx 1 f rtfr 9 )* / \ 

whence /0 = -_. == _ j, (i) 

#0 dx dy — dy d^x 



342. If, to determine <# we employ the relation 

• j- ^ 
sin = -=— , 

ds 

we shall obtain another formula for determining p. Thus, dif- 
ferentiating, we have 

, ,, ds d 2 y — dy d 2 s 
cos (f)d(p = J 2 , 

and, since cos^=-f-, 

ds 

i> _ds d*y — dy d 2 s < 
dx ds 

substituting this value of d(f>, we obtain 

_ dx ds 2 , v 

P ~ lls~d r y~~^dya r s' {l} 

Introducing derivatives with reference to s, we have, since 
d's 
d? = °> 

. dx 



<& 2 



360 APPLICATIONS TO PLANE CURVES. [Art. 343. 

343. In like manner, by employing 

dx 



COS (p = —r- 

ds 



to eliminate d<!>, wb derive 



_ dy ds* , x 



and, in terms of derivatives with reference to s, 

dy 
__ ds , v 

IF 

344. The expressions for p deduced below are employed in 
deriving certain results in Dynamics. 
We have 

ds cos <f> = <£r and afr sin fi = dy; 

whence, differentiating, 

d*s cos (f> — ds sin (f> d(f> = ^/ 2 ;tr, 
and </ 2 .y sin <f) + ds cos <j> d<l> = d*y. 

Squaring and adding, we obtain 

<VV) 2 + afr 2 ^ == {d'xf + (^ 2 j) 2 , 

whence . (^ + (jy-^ ; 

substituting — for ^, and solving for p, we have 



§ XXXIV.] EXPRESSIONS FOR p. 361 

By making s the independent variable, we obtain the deriva- 
tive expression 



ds'J ' \ds\ 



(2) 



The Radius of Curvature in Polar Coordinates, 



345. To obtain an expression for p in terms of r and (9, we 
have 

4 = t/>+ 6, whence = -$ = in ds , . . (1) 

1 dq> dO + dip y K) 

and tan tb = ~- ( 2 ) 

dr ' x J 

In differentiating equation (2) to obtain an expression for dip, 
dd may be regarded as constant ; since the result is to be ex- 
pressed in derivatives with reference to 6. 

Hence sec 2 $ dip = dr ~ ~ r f r d6, 

ds 
and, since sec ip = — , 
dr 

,. dr 1 — rd*r Jn 

dtp — — : dd. 

as 

Hence dd + d/> = ^ + ds* - nfr 



ds' dd > 



and, substituting in (1), we obtain 

ds* 



{dr 1 + ds 2 - rd-r) dd ' 
but ds" = dr- + r'dd\ 



362 



APPLICATIONS TO PLANE CURVES. [Art. 345. 



therefore 



or 



(dr* + rwy 



(2dr* + r"d& 2 -rd il r)dd y 
f dr 



d6 



+ T* 



r»+ 2 



(dry ^V 
dd* 



(3) 



\dd) 



346. To obtain p in terms of u f we eliminate r from the 
above equation thus: 



r= «- 



then 



dd 



1 du 

tf'dd 1 



and 



</V_ 2_ (du \ 2 _ 1 A 



dd 



dd" 



On substituting these values, we obtain 



P = 



11 V + m) 



(1) 



347. In the case of a polar curve, when there exists a simple 
relation between tp and 6, the expression 

ds 

may usually be employed with advantage. Thus the polar 
equation of the cardioid (Art. 271) is 



r — 4a sin 9 \e ; 



§ XXXI V.] EXPRESSIONS FOR p IN POLAR COORDIXA PES. 363 



whence [see equation (3) Art. 318] 



cot* = ^ = coti4 



tjr =$6 and f= 6 + tp = \ 9. 





Therefore p = — - = | — = } r cosec //• = § a sin 4r#. 



Relations between p, p, and r. 

348. In Fig. 75, if we denote (97? by / and /^ by r, we 
shall have 

p — rs'mt/-, and r=r cos if- . . (1) 

Now let Pmove along the curve at the rate — , 0^ 

then the tangent PR will rotate about P at the 

angular rate -=- , and OR will rotate about O at 

the same rate, since these lines are always at right angles to 
each other. The motion of the point R may be resolved into 
two motions ; one in the direction OR, and the other in the 
direction RP. Since the velocity of P in the direction OR is 
zero, the component of the velocity of R in this direction is 

r -— , while the component in the direction RP is p -=- . The 
dt' ' r dt 

first of these components is the rate of p, since is a fixed 

point ; therefore 

dp d$ . dp f . 

-4- — r — - whence r = -£- . . . (2) 
dt dt dt> v J 

The rate of r is the difference between the velocity of P in 



3°4 APPLICATIONS TO PLANE CURVES. [Art. 348. 

the direction RP, and the component velocity of R in the same 
direction ; therefore 

dr ds d6 . , , 

&=■*-*.-%* or * = *-***■ • ■ (3) 

349. By comparing the expressions for r in equations (1) 

dv 
and (2), and putting for cos ip its value -^— , we obtain 

d^'^'di'' 

whence p = -— = —- . . . .• . . . . (4) 

d(j) dp w 

An expression for p may also be derived from equation (3) ; 
thus, 

ds . dr 



or, by equation (2), 



p = W=*+df' (5) 



P=p+&. ....... (6) 



350. The point R in Fig. ^5 describes the pedal of the 
given curve, the polar coordinates of the pedal (Art. 327) being 

X = (j) — \n and r z — p. 

The ratio of the component velocities of R determines the 
direction in which R moves ; hence, denoting by i/> x the value of 
ip for this curve, we have 

P 
tan ipj. = — = tan tp. 



§ XXXIV.] RELATIONS BETWEEN p, /, AND T. 365 

Hence the angle between a curve and its radius vector at any 
point is equal to the angle between the pedal and its radius 
vector at the corresponding point. It is thence easily shown 
that the normal to the pedal bisects OP. 

351. From equation (2) Art. 348, we derive 

P P d(j> 2d4> ' 

this expression enables us to obtain the value of the product 
pt when/ is expressed in terms of the angle 0„ its inclination to 
the axis of x. Thus, in the case of the ellipse referred to its 
centre, we have 

p* = a 2 cos 2 #x + b 1 sin 2 # x * ; 
since dS x = d(j) we obtain, by differentiating, 

pr = — (a' - b') cos X sin 6 1 . 

352. The chord of curvature through the origin is that chord 
of the circle of curvature, corresponding to any given point P 
of a curve, which passes through P and through the origin. 
Denoting this chord by 2c, we have 

• , P 

2c = 2 f> sin tp = 2/Q - ; 

whence, putting -y- for - [see equation (4) Art. 349], 

,dr f s 

2c = 2 ?dt W 

* This equation, which may be found in works on Conic Sections, is equivalent 
to the polar equation of the pedal to the ellipse, when the centre is the pedal origin. 
(See Ex. XXXIII, 28.) 



366 



APPLICATIONS TO PLANE CURVES. [Art. 352. 



To derive an expression for 2c in terms of u, we have [equa- 
tion (2), Art. 321] 



f du\~ 
Jd) . 



-i 



Whence, taking logarithmic differentials, 

(u + ™)d» 



dp 

7" 



du 

dd 



du 
Substituting in equation (1), and putting for dr its value ¥ , 

we obtain 



W + 



2C = 2 



duY 

dd 



d %1 u 



(2) 



Examples XXXIV. 

1. Derive the expression 



from equations (2) Art. 342, and (2) Art. 343, by means of the rela- 
tion 



(!)'+ (I)'- 



2. Find the radius of curvature of the cycloid 

x — a (ip — sin ip) } y — a (1 — cos ^). 



§ XXXIV.] EXAMPLES. 367 

Prove tJiat — j (rr — ^), ««^ //^t' p = -rr . 

p = - 2 \\2ay). 

3. Find the radius of curvature of the involute of the circle, Art. 
301, 

# = # cos ?/' + # >/> sin </', _)' = sin ip — aip cos ?/\ 

p = a ^>. 

4. Find the radius of curvature of the parabola y — ^ax. 

^ (a + *)S 

5. Find the radius of curvature of the catenary 



= ;( f7+f ')• 



* 



and show that its numerical value equals that of the normal at the 
same point. 

f 

6. Find the radius of curvature of the semi-cubical parabola 

(4a + gxYx* 



ay 1 = x 3 



P 



6a 

7. Find the radius of curvature of the logarithmic curve 

y = as~' . 






368 APPLICATIONS TO PLANE CURVES. [Ex. XXXIV. 

8. Find the radius of curvature of the tractrix 



dx 


W -ft 




dy 


y 


p=^(^-/) i - 



9. Find the radius of curvature of the rectangular hyperbola 



xy = m 2 . 



2/ra 2 



10. Find the radius of curvature of the hyperbola 



a* '¥~ 1 ' 

(e*x* - a*)* 

n. Find the radius of curvature of the parabola of the nth degree 

(x* + n*f)% 



a n ~ x y = x n 



P 



n (n — 1 )xy ' 



12. Find the radius of curvature of the cissoid 

2. 
x* 



(2a — x)^ 

p = aVx(Sa — $x)% 
3(20 — xy 

13. Given the curve y* 4- x % 4- a(x' 2 + y^) — a*y. 
Find the value of p at the origin. See Art. 338. 



§ XXXIV.] EXAMPLES. 369 

14. Given the curve ay = bx % + cxy. 
Determine the value of p Q . 

Po =00 - 

15. Given the parabola a n ~ x y = x n . 
Determine the value of p Q . 

For n < 2 we have p = o ; 

" » > 2 " p = &. 

16. Given the curve ax 3 — 2#\ry + cy % = x* + y. 

Determine the values of p . 

I? 2 
For the branch tangent to the axis of x, p o = — ; 

72 

for the branch tangent to the axis of y, p Q =— . 

17. Given the curve x 4 — ^axy- + 2af — o. 

Find the value of the radius of curvature at the cusp. See Fig. 
20, Art. 215. 

Po = O. 

18. Given the curve x* — axy -f axf + \a^f — o. 
Determine the value of p Q . See Ex. XXVIII, 24. 

Po = i*. 

19. Show by means of the expression for p o that the curve 

x* — axy — axy 1 -f a~y~ = o 
has an isolated point at the origin. See Ex. XXVIII, 23. 

20. Find the radius of curvature at the origin, the equation of the 
curve being [see Ex. XXVIII, 25] 

x — f axy — axy + ay = o. 

p o = a, and p = \a. 



370 APPLICATIONS TO PLANE CURVES. [Ex. XXXIV. 

21. Given the curve x b — ^ay* -f 2ax % y + # 2 .ry 2 = o. 
Find the values of p . See Ex. XXVIII, 26. 

p — — i-a, at the cusp ; 

p Q = ^<z, for the other branch. 

22. Given the curve a (y 2 — .# 2 ) = x s . 
Determine the values of p o . 

Turning the axes forward 45 °, the equation of the curve becomes 

4i/2-axy = (x—y)\ 

p Q = 2a Y2, and p Q = — 2a V2. 

23. If -f- •= — at every point of a curve, find the value of p. 

Solution : — 

a 

, -ia . d(f) s 2 

<p = tan - , whence -j-= 5- ; 

s ds a 

1 r fl * + ^ 

therefore p = . 

24. Find the radius of curvature of the parabola 

V* + Vy — 2 V<*> 

_(x + y)% 

P ~ Va ' 

25. Find the radius of curvature of the cubical parabola 

a 2 y — x s . 

_ (a' + 9 x*)* 
9 ~ 6a*x 

26. Find the radius of curvature of the prolate cycloid 

x = aip — b sin ip f y = a — b cos ip. 

_ (a 2 + b 2 — 2abcos*J:)* 
' b(a cos ip — b) 



§ XXXIV.] EXAMPLES. 37* 



27. Find the radius of curvature of the three-cusped hypoeycloid 

x = b (2 cos //' + cos 21/'), y — b{2 sin tp — sin 2t/-). 

p = — Sb sin I?/'. 

28. Find the radius of curvature of the epicycloid 

t i\ , 7 a + b . . fX . , , . # + £ . 

* = (# + 0) cos y; — b cos — — ip, yz=(a + b) sin >/; — b sin —7- '/'. 

4/; (a + b) . ail) 
p = v — j 1 - sin — 7- . 
7 a + 2^ 2^ 



29. Find the radius of curvature of the curve 

x = 2<r sin 2^ cos 2 ^, _y = 2c cos 2^ sin* ^. 

p = ac cos 3^. 

30. Find a general expression for the radius of curvature of any 
conic referred to its focus, the equation being 

1 — e cose 

a (1 — e 2 ) (1 — 2c cos + e*)'* 

p = — — 



(1 — e cose) 3 

31. Find the radius of curvature of the trisectrix 

r = 2a cos — a. 

a (5 — 4 cose)* 
9—6 cos e 

32. Find the radius of curvature of the lemniscata 

r = a 2 cos 26. 
See Art. 347. p = — 

3' 



37 2 APPLICATIONS TO PLANE CURVES. [Ex. XXXIV. 

$$. Find the radius of curvature of the parabola whose equation is 
r cos 2 -|9 = a, the focus being the pole. 

34. Find the radius of curvature of the equilateral hyperbola 



r 

35. Find the radius of curvature of the equiangular spiral 

r — as n0 . 

p = rV(i + O- 

$6. Find the radius of curvature of the lituus 

r*B = a\ 

_ r (4a 4 + r*)* 
P ~2a*{ 4 a*-r i )- 

37. Given r m — a m cos md, prove that r m + 1 = a m p, and thence de- 
rive the value of p by means of equation (4) Art. 349. 

r 2 

^ ~~ (m + 1 )p ' 

38. Show that the chord of curvature through the origin, in the 

Ar 

case of the cardioid r = 2a (1— cos 6), is — . 

3 



§ XXXV.] BVOLUTES. 373 

XXXV. 
E volutes. 

353. The cvolutc of a curve is the locus of its centre of curva- 
ture. 

Since, when the point P, Fig. j6, moves along the given 
curve, the motion of the centre of curvature has no component 
perpendicular to the normal PC (see Art. 332), this line is tan- 
gent at C to the curve described by the point C\ consequently 
the evolute may be defined as the curve whose tangents are the 
7iorma/s to the given curve. 

Now, since the point P moves in a direction perpendicular 
to PC while C has no motion except in the direction PC; if we 
regard P as a fixed origin on the line PC, the rate of the motion 
of C along this line will be identical with its rate along the arc 
of the evolute. Hence PC may be regarded as a tangent line 
rolling upon the evolute, while the fixed point P of this tangent 
describes the original curve. 

A curve generated by any fixed point on a rolling tangent 
is called an involute. Thus the original curve is one of the in- 
volutes to its own evolute, and may be traced by the extremity 
of a cord which is being wound upon the evolute or unwound 
from it. 

354. We proceed to the methods of deriving the equation 
of the evolute. 

To obtain expressions for the coordinates (%\y') 
of the centre of curvature C (Fig. j6) in terms of 
x and y, we project the line p on the coordinate 
axes. The inclination of PC to the axis of x is 
^ + 90 ; hence the projections are Fig. 76. 




374 APPLICATIONS TO PLANE CURVES. [Art. 354. 

x — x = — p sin <p — — P-^, (1) 

and y' — y = p cos (f> = p —-. ..... (2) 

Substituting for p in the above equations the value of p 
given in equation (1), Art. 335, we have 



x'-x = ■__•-£, ( 3 ) 



d' z y 
dx" 


dy 

dx' * ' • ' ' 


• - m 




dy 





From these equations j and its derivatives must be elimi- 
nated by means of the equation of the given curve. The rela- 
tion between x' and y found by subsequently eliminating x 
from the two equations is the equation of the evolute. It should 
be remarked however that the latter elimination is frequently 
impracticable.* 

355. As an illustration, let it be required to find the evolute 
of the common parabola 

y = 2a*x 2 ; 

whence s=S)* 



* Another method of obtaining the equation of the evolute will be found in 
Section XXXVII. 






§ XXXV.] EVOLUTES. 375 

and -4 = - — - 

dx- 2x i 

By substituting in formulas (3) and (4), Art. 354, these values of 

dy . d 2 y 

y, -y- y and -~, we obtain 
dx dx" 

x = 2a + ix, (1) 

and y = --T-; (2) 

eliminating ^r, we have 

27#/ 2 = 4 (V — 2#) 3 , 

the equation of the evolute, which is, therefore, a semi-cubical 
parabola having its cusp at the point {2a, o). 

356. When it is more convenient to regard x as a function 
of 7, we employ the equations 

, -(f)" 
"-'=—&- <■> 

1 + (— V 

t , \dy J dx , N 

and y — y — — — — ^— . — - (2) 

' d 2 x dy ' 

These equations are obtained by substituting in equations 
(1) and (2) of Art. 354 the value of p derived in Art. 340. 

It is sometimes advantageous to use one of these formulas 



37^ APPLICATIONS TO PLANE CURVES. [Art. 356. 

in connection with one of the formulas of Art. 354. For ex- 
ample, to find the evolute of the hyperbola 

~2 £2 ~ l » 



we have y == ± - ^(V — # 2 ), 



whence -f_ = ± £ 



dx a V(x 2 — <f) 

j d*y _ ab 
and -/j = q= — I . 

^ (;r 2 - c?)* 

Substituting in equation (3) Art. 354, and reducing, we have 

._ *>& + ?) 
X ~ a* * 

In a similar manner we obtain, by means of equation (2) above, 

_ yp + v) 

y ~ v 

Eliminating x and y from the equation of the hyperbola by 
means of these results, we derive for the equation of the 
evolute 

(axf - (by'f = (a 2 + b*)k 

ds 
357. By substituting -yy for p in equations (1) and (2) of 

Art. 354, we obtain the formulas 

, dy , , dx f x 

x - x= -ii,' and y ~ y = w • • • (I) 



§ XXXV.] EVOLUTES. 377 

which may frequently be employed with advantage. For ex- 
ample, in the case of the cycloid, we have 

x — a (//- — sin ?/•), y = a (i — cos f/>). ... (2) 
Differentiating, 

dx = a (1 — cos ip) dip, and dy = a sin ^ aty? ; 

. dy sin 

whence -f- = ! = cot - = tan 6, 

dx 1 — cos if) 2 

therefore (f> = \n — J^, and dcf>= — hlip. 

Substituting in formulas (1), we obtain 

x' — x = 2a sin ip, y' — y = — 2a{\ — cos */-) ; 

and, eliminating x and y by equations (2), we have the equa- 
tions of the evolute 

x — a(rp + sin ip), y — — a (1 — cos //•). 

If we transfer the origin to the point {—an, — 2d), and 
denote by ;tr and j the coordinates with reference to the new 
origin, we shall have 

x = x + an and y = y + 2a ; 

whence, denoting ?/? + tt by ^-', we obtain the equations 

x — a(rp' — sin ip'), y — a(\ — cos xp'), 

which are identical in form with equations (1). 

Therefore, the evolute of the cycloid is an equal cycloid situ- 
ated below the axis of x, and having its vertex coincident 
with a cusp of the given cycloid. 



37 8 APPLICATIONS TO PLANE CURVES. [Art. 358. 



The Length of an Arc of the E volute. 

358. It follows from the mode in which a curve may be 
generated from its evolute, as explained in Art. 353, that any 
continuous arc of the evolute is equal to the difference between the 
corresponding values of p. 

For example, the values of p corresponding to the vertex 
and to the cusp of the evolute determined in the preceding 
article are respectively zero and 4a ; hence, the length of one 
half of the evolute is 4a. It follows therefore, since the evolute 
is a cycloid equal to the given cycloid, that the length of an 
entire branch of the cycloid is eight times the radius of the 
generating circle. 

359. It is evident that the point of the evolute which cor- 
responds to a maximum or to a minimum value of p will be a 
cusp. Thus, since the radius of curvature of an ellipse is a 
minimum at each extremity of the major axis, and a maximum 
at each extremity of the minor axis, the evolute of this curve 
consists of four branches separated by cusps. 

The entire length of a curve of this character can only be 
found by determining separately the length of each branch. 



Involutes and Parallel Curves. 

360. The method of generating involutes given in Art. 353 
shows that the involutes of a given curve cut its tangents at 
right angles ; in other words, they are trajectories (see Art. 282) 
of these tangents, and since the constant angle is a right angle 
they are said to be orthogonal trajectories. Hence any two of 
these involutes are curves having common normals, and it is 
evident that the portion of the normal intercepted between 
them is constant. Curves thus related are called parallel curves. 



§ XXXV.] THE LENGTH OF AN ARC OF THE EVOLUTE. 3/9 

To find the general equation of the involute of a given 
curve requires the aid of the Integral Calculus; but when one 
of the involutes is known, the problem is reduced to that of 
finding the general equation of the parallels to this curve. 

If PP = c be the constant intercept on the normal, taken in 
the direction fi + 90 , we have, by projecting PP upon the 
axes, 

x' — x = — c sin (j) ) 
y ' — y — c cos $ ) ' 

in which (x, y) is a point on the known involute, <j> the inclina- 
tion of the curve at this point, and (x', y') the corresponding 
point on the parallel involute. 



The Radius of Curvature at a Cusp and at a 
Point of Inflexion. 

361. Since 

ds 

p will change sign whenever ds or d(j> changes sign, unless 
these differentials change sign simultaneously. 

When ds changes sign the generating point reverses the 
direction of its motion abruptly, and forms a cusp ; for this 
reason a cusp is sometimes called a stationary point. 

When d(f) changes sign the tangent at the generating point 
changes its direction of rotation, and, if ds does not change 
sign simultaneously, we have a point of inflexion. The tangent 
at a point of inflexion is, therefore, sometimes called a stationary 
tangent. 

At an ordinary or ceratoid cusp, at which the two branches 
of the curve lie on opposite sides of the common tangent, it is 
easy to see that the tangent does not change its direction of 



380 APPLICATIONS TO PLANE CURVES. [Art. 36 1. 

rotation as the generating point passes through the cusp ; that 
is, d<j> does not change sign. At a ramphoid cusp, however, 
both branches of the curve lying on the same side of the tan- 
gent, it is evident that the direction of rotation of the tangent 
line is reversed as the generating point passes through the cusp ; 
in other words, ds and d$ change sign simultaneously. 

362. It follows from the preceding article that p changes 
sign at a ceratoid cusp and at a point of inflexion ; hence at 
these points the value of p must be either zero or infinity, and 
consequently the evolute must either pass through the point in 
question or have an infinite branch to which the normal at this 
point is an asymptote. 

At a ramphoid cusp, however, p does not change sign, and 

. . , ds ' . , . . o 00 . 

since its value -— takes an indeterminate form, - or — , it may 
d<p o co 

have a finite value which is of course the same for both branches 

of the curve. See Ex. XXXIV, 18, 21. 



The Radius of Curvature of the Evolute, 

363. Denoting by p' the radius of curvature of the evolute, 
we have 

, ds f s 

p =i$ (I) 

Now it follows, from Art. 358, that the rate of s' is identical 
with that of p, hence 

ds' = dp ; 

and, since the tangent to the evolute is the normal to the given 
curve, 

(j> f =z (j) + J7r, whence d</>'— d<f>. 



§ XXXV.] RADIUS OF CURVATURE OF THE EVOLUTE. 381 

Substituting in (1), \vc have 






'-% ^ 

and substituting in (2) the value of d$ from the equation 

ds 
P = ^' 

dp . x 

P = ^ ; <3) 

also, by eliminating p from equation (2), we obtain 

P ~W w 

364. When in the case of a given curve the relation be- 
tween p and $ is readily obtained, equation (2) enables us to 
express p' in terms of (p\ and the relation thus found sometimes 
serves to determine the evolute. 

For example, in the case of the cardioid 

r — 4a sin J0, (1) 

it was shown in Art. 347 that <f> = §6> and that p = — sinJ0, 

hence p — — sin-^ (2) 



Therefore p'— -jj = — cos J^, 



^p 8# 
d$ 



but since <f> = <j>' — %n, cos \(ft = sin (£zr + -J0) = sin \ (ft + n), 
hence p — — sin-J(^'+ n) (3) 



382 APPLICATIONS TO PLANE CURVES. [Art. 364. 

Comparing equations (2) and (3) it is evident that the evo- 
lute is a cardioid whose axis is one-third of that of the given 
cardioid, and, since the radius of curvature is zero at the cusp 
of the given cardioid, the vertex of the evolute coincides with 
this point. 

The length of the evolute may be found by the method of 
Art. 358. For the semi-perimeter of the evolute is equal to the 
value of p at the vertex of the given cardioid, but •by equation 

o 

(2) this value is — a, or twice the axis of the evolute. Hence the 

3 

perimeter of a cardioid is four times its axis. 



Examples XXXV. 

1. Find the equation of the evolute of the ellipse. 

(ax')* + (fy') i = {a'-b^ 

2. Show that the entire length of the evolute of the ellipse is 

fa" b" 
4 {j~a 
See Art. 359. 

3. The equation of the equilateral hyperbola being 

xy = m*, 

prove that 

, mfm x\ s . , , mfm x\ 3 

x+y = —[ 1 ), and x—y = —[ ; 

2 \x mj 2 \x mj 

ind thence derive the equation of the evolute. 

(x + _/) J — (x'—y'Y 3 — (4^)3. 

4. The equation of the semi-cubical parabola being 

3 2 

x = ay , 



§XXXV.] EXAMPLES. 3*3 

prove that 

| *--*-£• and ^=^ + j)l/f: 

and thence derive the equation of the evolute. 

729 ay* — i6[2a 4- V{a 2 — i&ax')]' [ tf(a* — iSax') — a]. 

5. Given the equation of the tractrix (see Art. 303) 



prove that 



a + v(# 2 — y-) ., 8 ,, 

* = a log - - ^ - V(a- -f) ; 



r a 1 , » , # + y(tf 2 —J' 2 ) 
y = — , and x = a log * ^^ > 

7 7 



and deduce the equation of the evolute. 

x' x' 

6. Given the equation of the catenary 

X X 

prove that 

X X 

y'=2y, and x' — x — — f s a — e a \ 
and deduce the equation of the evolute. 

X ' = a 1 /±(y -*»')* T I (ys _ 4al i. 

2a 4a 

7. Find the equation of the evolute of the four-cusped hypocycloid 

x% + yl = <A 

Expressing -f- and — 2 in terms of jc and y, we obtain 



384 APPLICATIONS TO PLANE CURVES. [Ex. XXXV. 

x'= x + 3 (xy*)^j and y'—y + 3 {x*y)*, 
and thence 

x'-\-y= (x :i +y 3 ) 3 , and x — y'= (x s — y*) a . 
Whence the required equation is (x -\- y)* + (x —y')^ = 20 s . 

8. Find the equation of the evolute of the curve 

y=(2b — a) cos ip — {b — a) cos 3 ip, x — (2a — b) simp — (a — b) sin 3 ip. 

Employ the method illustrated in Art. 357. 

**+/*= [ 2 (^-^)]l. 

9. Find the equations of the evolute of the curve 

x = c sin 2^(1 + cos 2iy), y = c cos 2^(1 — cos 2ip). 

x'= 2c{— 2 sin ip cos 3^ + sin 2ipcos^ip); 
y — 20(2 cosip cos 3^ + cos 2^ sin 2 ^?). 

10. Find the cusps and asymptotes of the evolute of the lemniscata 

r 2 — a? cos 20. 
See Art. 359 and Art. 362. 

11. Prove, by the method of Art. 364, that the evolute of the loga- 
rithmic spiral, 

r = a£ ne , 

is a similar logarithmic spiral, and show that the constant ratio be- 
tween the corresponding radii of curvature is equal to n. 

12. Prove that the evolute of an epicycloid having a cusp on the axis 
of x, the radii of the fixed and rolling circles being a and b respec- 
tively, is a similar epicycloid having its vertex on the axis of x y and 
whose radii are 

a' ab 

a — — — and b = — ■ — - . 
a +2b a + 2b 



§ XXXV.] EXAMPLES. 385 

13. Find the radius of curvature, and the equation of the evolute of 
t cissoid 



the cissoid 

,2 — x " 

2a — x 



P 



S(2a -"xY~ ' 4 ° 96 ^ * + II S 2a 'y°' + 2 7/ 4 = o. 

14. Show, by means of the result obtained in Example 12, that one 
of the involutes of the epicycloid, the radii of whose fixed and rolling 
circles are a and b, is the similar epicycloid whose radii are a + 2b and 
b (a + 2b) 

; and thence derive the general equation of the involutes. 

See Art. 360. 

__(*_+ 2b )( a ± V) nne ,.b(a+ 2b) a + b . a + 2 b , 

■* — cos ip + — cos — r — ib — c cos ib: 

a a b T 2b T 

(a-r 2b)(a + b) b {a 4- 2b) . a + b . . a + 2 b , 

y — sin ip + sin — : — ib — c sin r— ib. 

a a b ' 2b r 

15. Find the general equations of the involutes of the cycloid. 
See Art. 360. 

x — a (n + if) — sin >/:) — c cos \ip\ 
y'= a {3 — cos ip) + c sin £#. 



XXXVI. 

Envelopes. 



365. The curves determined by an equation involving x 
and y together with constants to which arbitrary values may 
be assigned are said to constitute a system of curves. The 
arbitrary constants are called parameters. When but one of 



386 _ APPLICATIONS TO PLANE CURVES. [Art. 365. 

the parameters is regarded as variable, denoting it by a, the 
general equation of the system of curves may be expressed thus : 

f(x,y, a)=o. ...:... (1) 

When the curves of a system mutually intersect (the intersec- 
tions not being fixed points), there usually exists a curve 
which touches each curve of the system obtained by causing the 
value of atmto vary. 

For example, the ellipses whose axes are fixed in position, 
and whose semi-axes have a constant sum, constitute such a 
system ; and, if we regard the ellipse as varying continuously 
from the position in which one semi-axis is zero to that in which 
the other is zero, it is evident that the boundary of that por- 
tion of the plane which is swept over by the perimeter of the 
varying ellipse is a curve to which the ellipse is tangent in all 
its positions. A curve having this relation to a given system 
of curves is called the envelope of the system. 

Every point on an envelope may be regarded as the limit- 
ing position of the point of intersection of two members of the 
given system of curves, when the difference between the cor- 
responding values of a is indefinitely diminished. For this 
reason, the envelope is sometimes called the locus of the ultimate 
intersections of the curves of the given system. 

366. If we differentiate equation (1) of the preceding arti- 
cle (regarding aasa variable as well as x and y) the resulting 
equation will be of the general form 

/; (x, y, a) dx + f y {x, y, a) dy + f' a {x, y, a) da = O.* . (2) 

In this equation each term may be separately obtained by 
differentiating the given equation on the supposition that the 
quantity indicated by the subscript is alone variable. See 
Art. 64. 



XXXVI.] ENVELOPES. 3S7 



From equation (2) we derive 

<ty = _ f, (*> J'> *) _ /„' (•*•>.?, <* ) da 



(3) 



In Fig. 77 let PC be the curve corresponding to a particular 

value of a, and let P be the point (x, y) ; then 

dy 
the expression for -y- given in equation (3) 



determines the direction in which the point P 
is actually moving when x, y, and a vary 
simultaneously. This direction depends there- fig. 77. 



S. 




ia 



fore in part upon the arbitrary value given to the ratio — . 

ax 

367. Now if a were constant da would vanish, and equa- 
tion (3) would become 

dy _ f'Js,y,«) 

dx f y (x,y,a) W 

dy 
This expression for -~- determines the direction in which P 

moves when PC is a fixed curve. 

Let AB be an arc of the envelope, and let C be its point of 
contact with PC. Now, if P be placed at the point C, it is 
obvious that it can move only in the direction of the common 
tangent at C, whether a be fixed or variable. It follows there- 
fore that, at every point at which a curve belonging to the 

system touches the envelope, the expressions for -f- given in 

dx 

equations (3) and (4) must be identical in value. 

Assuming that f' x (x, y, a) and /' (x, y, a) do not become in- 
finite for any finite values of x and y, the above condition re- 
quires that 

f a (x,y,a) = (5) 



388 APPLICATIONS TO PLANE CURVES. [Art. 367. 

Hence the coordinates of every point of the envelope must 
satisfy simultaneously equations (5) and (1) ; the equation of 
the envelope is therefore obtained by eliminating a between 
these two equations. 

368. Let it be required to find the envelope of the circles 
having for diameters the double ordinates of the parabola 

y 1 = 4ax. 

If we denote by a the abscissa of the centre of the variable 
circle, its radius will be the ordinate of the point on the para- 
bola of which a is the abscissa, the equation of the circle will 
therefore be 

f- + (x — ocf — 4aa = (1) 

Differentiating with reference to the variable parameter a, we 
have 

— 2 (x — a) — 4a = o, 

or a = 2a + x ; (2) 

substituting in (1), and reducing, we obtain 

y = 4a(a + x) (3) 

The envelope is, therefore, a parabola equal to the given para- 
bola and having its focus at the vertex of the given parabola. 

369. In the above example every point of the envelope is 
a point of contact with a member of the given system of circles ; 
for the value of a corresponding to any given point of the 
envelope is always real, since it is determined by means of 
equation (2). This equation serves also to determine the 
abscissa of the points corresponding to a given value of a ; 
but, when a is less than a, the ordinates of these points, de- 
termined by equation (3), are found to be imaginary. Hence 
a portion of the system of curves given by equation (1) does 






§ XXXVI.] ENVELOPl 389 

not admit of an envelope ; in fact, the circles belonging to this 
part of the system do not intersect. 

On the other hand, examples are sometimes met with in 
which the value of a corresponding to certain real points on 
the envelope is found to be imaginary. When this is the case, 
the envelope contains an arc or arcs which do not properly 
belong to it as geometrically defined. 

Fixed Points of Intersection, 

370. When the equation of a system of curves is in a 
rational integral form with respect to ,r, y f and or, the order of 
the system is indicated by the $legree of the equation with re- 
spect to x and y, and the index of the system by its degree 
with reference to a. 

The equation of a system whose index is unity may be 
written in the form 

F t (x,y) + *F,{x,y) = o (1) 

Every point which satisfies the simultaneous equations 

fi (*i y) = ° and f% (*i .r) = °. • • • ( 2 ) 

evidently satisfies equation (1) for all values of a: hence all the 
curves of the given system pass through the fixed points in 
which the curves (2) intersect. 

Moreover, two curves of the given system cannot intersect 
in any other points ; for, at a point common to any two curves 
of the system, we have 

F T (x, y) + a x F % (x, /) =0, (3) 

and F t (x, y) + a, F, f.r, y) = o (4) 

By subtracting we obtain 

(a\ — a 9 ) F, ('.r, y) = o, 



390 APPLICATIONS TO PLANE CURVES. [Art. 370. 

F^(x, y) — and consequently F x (x, y) = o ; 

hence every intersection of the curves (3) and (4) coincides 
with one of the fixed points determined by equations (2). 

A general equation of the form (1), therefore, represents a 
system of curves which intersect only in fixed points and con- 
sequently does not admit of an envelope. Such a system is 
called 2. pencil oi curves. 

371. If the equation of a system of curves can be put in the 
form 

^1 (*, y) + /{*, J, a)-F a (x, y) = o, . . . . (1) 

all the curves will pass through the intersections of 

F T (x, y) — o and F 2 (x, y) = o ; . . . . (2) 

but in this case the curves of the system may also intersect in 
other points, and hence the system may have an envelope. 

By differentiating equation (1) with reference to a we 
derive 

fL (*« y> a )' F * (*> y) = °i (3) 

which is satisfied by F i (x,y) = o t . . (4) 

and also by f a { x yy> a ) = ° (5) 

Equation (4) indicates the existence of the fixed points of 
intersection, and equation (5) determines the envelope. 

372. For example, let it be required to find the envelope of 
the series of parabolas obtained by varying the angle of elevation 
of a projectile having a given initial velocity. 

The equation of the path of a projectile, when no allowance 
is made for the resistance of the air, is 

y = x tan a =- , 

4/z cos a 



§ XXXVL] FIXED POINTS OF INTERSECTION. 391 

the origin being at the point of projection, and a denoting the 
angle of elevation above the horizontal plane. Putting a in 
place of tan or, we obtain 

4/1 (y — ax) + (i + a r ) X s = o, 

or 4/iy + x* + x (xa- — 4/1 a) = (i) 

Taking the derivative, we have 

2x (xa — 2k) = (2) 

The solution „r = o indicates the existence of fixed points of 
intersection in the system of curves. The equation F z (x f y) = o 
is, in this case, 

4A7 + **=<> (3) 

The fixed points of intersection are therefore the points 
common to the line x = o, and the curve given by equation (3). 
These points are the origin and the point at infinity on the 
parabola 4/iy + x 1 = o. 

The other solution of equation (2), viz., 

xa — 2/1 — o, (4) 

determines the envelope. Substituting in equation (1) the value 
of a obtained from equation (4), we derive 

x' = Ali{h-y) (5) 

for the equation of the envelope, which is, therefore, a parabola 
with its axis vertical and its focus at the origin. 



Equations in an Irrational Form. 

373. It is assumed in Art. 367 that f' x and f cannot be in- 
finite for finite values of x and y. Cases in which these func- 
tions become infinite cannot arise when the given equation is 



39 2 APPLICATIONS TO PLANE CURVES. [Art. 373. 

rational in form ; but when an irrational form of the given equa- 
tion is employed, a relation between x and y can sometimes be 
found, which gives 

f' x (x, y, a ) = 00 and /' (x, y, a) = 00, 



and does not at the same time give an infinite value to f' a {x,y, <*). 

The value of -f- det< 
ax 

in such cases, reduce to 



The value of -J- determined by equation (3), Art. 366, will, 



f y (*> y> <*) 

which is identical with the value given by equation (4), Art. 
367 : hence the points thus determined belong to the enve- 
lope. 

374. For example, if the given equation is 

y — 2ax ± y(y* — ^ax) =0, (i) 

we have 

/ = I ± ,/ / -n and / =-2«T —rr—* x- 

y V(y"-4ax) Jx Y(f — 4a*) 

These expressions are both infinite when 

y 2 — 4ax = 0, ....... (2) 

while f is finite. Hence equation (2) (since it does not con- 
tain a) is the equation of an envelope. 

375. By putting/^ = o in this example, we obtain 

x — o\ 



§ XXXVI.] EQUATIONS IX AX IRRATIONAL FORM. 393 

this line does not, however, properly belong to the envelope, 
as will appear from the rationalized form of equation (i), viz., 

ocx" — ixxy + ax = o, 

the locus of which is composed of the two straight lines 

x = O and y — ax =0. 

a 

Since the first of these equations is independent of a it con- 
stitutes a fixed branch of the given system. 

Whenever the variable system of lines contains a fixed 
branch, the motion of a point situated on this branch will be 
independent of a ; hence such points satisfy the condition em- 
ployed in Art. 367 to determine an envelope. 

The envelope of the variable line 

a 

y — ax = 

a 

(obtained from this form of the equation by the condition 
fi ~ °) * s °f course the parabola whose equation is found above. 



Two Variable Parameters. 

376. When the equation of the given curve contains two 
variable parameters connected by an equation, only one of 
these parameters can be regarded as arbitrary, since, by means 
of the equation connecting them, one of the parameters can 
be eliminated. Instead, however, of eliminating one of the 
parameters at once, it is often better to proceed as in the fol- 
lowing example. 

Required, to find the envelope of a varying circle whose centre 
moves 011 a given circle, and whose circumference is always tangent 
to a fixed diameter of the given circle. 

Denoting the coordinates of the centre of the varying circle 



394 APPLICATIONS TO PLANE CURVES. [Art. 376. 

by a and /?, the centre of the fixed circle being the origin, and 
the fixed diameter being the axis of x, the equation of the vari- 
able circle is (x — af + (y — /3) 2 = /5 2 , 

or (x — af + y* — 2py =0 (1) 

Denoting the radius of the fixed circle by a we have also 

a* + /3* = a*; ....... (2) 

differentiating (1), we have 

(x — a) da + yd/3 = 0, (3) 

and from (2) ada + fid ft = o (4) 

We have now four equations from which to eliminate a, fi, and 

doc 
the ratio -^ . Transposing, and dividing (3) by (4) to eliminate 
dp 

da 

-j- , we have 

d P x-a _ y m , . 

a ~fi' {b) 

substituting in equation (1) the value of x — a, 



,2„,2 



zr+y = 2 fiy, 



or, by equation (2) ~ = 2f3y. 

P 

Whence y — o, and \c£y = fi 3 (6) 

y = o is the equation of the diameter of the fixed circle, which 
evidently constitutes a part of the envelope. 
From equation (6), we have 

2 3 



§ XXXVI.] TWO VARIABLE PARAMETERS. 395 



and, from equation r 

xB (fix 



y + ft aS + 2^ ' 
Substituting these values of a and /3 in equation (2), 

4- yi - * 



(a :i + 2 3 ^ 3 ) 2- 

whence, clearing of fractions and reducing, 

4< (V +y 2 -d) = icfcjfty 
or 4 (x- + y — a")* = 2?ay\ 

This is the equation of the tzuo-cusped epicycloid. See Art. 294. 

The Application of Undetermined Multipliers to 
Envelopes. 

311. In some cases the elimination of the parameters is 
facilitated by the application of the method of undetermined 
multipliers as illustrated in the following example. 

A variable circle moves with its centre on the circumference 
of a fixed circle, while its radius has a given ratio to tlie chord 
joining its centre to a fixed point of the circumference. 

Taking the fixed point as an origin, and a diameter of the 
circle as the axis of x> the equation of the fixed circle is 

x 1 + j/ 2 — 2ax — o. 

Let a and (3 denote the coordinates of the centre of the vari- 
able circle, and n the given ratio of its radius to the chord. 
The length of the chord being |/(a' 2 + /?-), we have, for the 
equation of the variable circle, 

(x - of +(y- pf = 1? («« + (5% 



39^ APPLICATIONS TO PLANE CURVES. [Art. 377. 

and, for the relation between the parameters, 

a 1 + fi 2 — 2aa = O (i) 

By substituting the value of a* -\- fi 2 from equation (1), the equa- 
tion of the variable circle becomes 

x 1 + y- — 2ax — 2fiy — 2 (?z 2 — 1) aa = o. . . (2) 

Hence the differential equations are 

(a — a) da + fidfi — O, 

and [> + (/z 2 — 1) #] da + ^/? = o. 

Multiplying the second by A, and subtracting from the first, 

[a — a — A {x + (/r — 1) a\\ da + {(3 — \y) dfi = o. 

Since A is arbitrary, we are at liberty to assume the coefficient 
of da to be zero, and, as a consequence of this assumption, we 
have the coefficient of df3 likewise equal to zero ; whence 

a — a — A [x + (ri 2 — 1) a] = O, . . . . (3) 

and /J — Aj = o (4) 

We have now to eliminate a, ft, and A between equations 
(1), (2), (3), and (4). To effect this elimination, we multiply 
(3) by a, (4) by fi, and add, thus obtaining 

« 2 + /3- — aa — A (ax -f fiy + (V — 1) #<*) = O. 

Simplifying this equation by means of (1) and (2), we obtain 

aa-il(x*+f)=o; ....... (5) 

eliminating a between (3) and (5), 

i — 2 ^ 2 

~~ x* + / — 2#.r — 2 (;r' — 1) c? ' 



§ XXXVI.] TWO VARIABLE PARAMETERS. 397 

whence, from equation (5), 

a (x 2 + f) 



a — 



x 2 +y* - 2<xx — 2 (?i 2 — 1) a 2 9 
and, from equation (4), 



x 2 + y 2 — 2ax — 2 (u 2 — 1 ) a' 2 

By substituting these values of a and (5 in equation (2), we 
derive 

< . o _ 9 ** (^ + /) 4- (^ - 1) ^ 2 (* 8 + /) + 2^y _ 

which reduces to 

(x 2 + /) 2 - 4ax {x- + /) - 4 (t* 2 - 1) a 2 (^ 2 + j 2 ) - 4^> 2 = o, 
or (x 2 + y) 2 — (4^^ + 472 V) (V + y-) + 40V = o. 

The envelope is therefore a limacon. See Art. 269. 

Examples XXXVI. 

1. Find the envelope of the system of parabolas represented by the 
equation 

in which a is an arbitrary parameter and c a fixed constant. 

2. Find the envelope of the circles described on the double" ordi- 
nates of an ellipse as diameters. 

„' + f ^ fi> '• 



39$ APPLICATIONS TO PLANE CURVES. [Ex. XXXVI. 

3. Find the envelope of the ellipses, the product of whose semi- 
axes is equal to the constant c 1 . 

The conjugate hyperbolas, 2xy = ± c 1 . 

4. Find the envelope of a perpendicular to the normal to the para- 
bola, y- — 4ax, drawn through the intersection of the normal with the 

axis. 

y 1 = 4a (2a — x). 

5. Find the envelope of the ellipses whose axes are fixed in posi- 
tion, and whose semi-axes have a constant sum c. 

The four-cusped hypocycloid, x* + y Ti = c*. 

6. A circle moves with its centre on a parabola whose equation is 
y = ^ax, and passes through the vertex of the parabola ; find the en- 
velope. 

The cissoid, y 1 (x + 20) + x s = o. 

7. A straight line cuts the coordinate axes in such a manner that 
the product of the intercepts is constant and equal to c 1 ; find the en- 
velope. 

xy — J<r 2 . 

8. A perpendicular to the tangent to a parabola is drawn at the 
point where the tangent cuts the fixed line x — c ; find the equation 
of the envelope of this perpendicular. 

The parabola, y* = — 4c (x — a — c). 

9. Prove that the envelope of the system of curves 

y — ax ± V[f{x,y)] =0 is / (x } y) = o. 
10. Show that the hyperbolas given by the equation 

<* 

— \- — — 1, when a + p = a 
x y 

form a pencil, and therefore do not admit of an envelope. 



§ XXXVI.] EXAMPLES. 399 



ii. The centre of an hyperbola passing through the origin, and hav- 
ing asymptotes parallel to the axes moves upon the circle x" -f y 1 = a 1 ; 
find the envelope. 

x 1 / = a: (a 2 + /). 

12. A straight line of fixed length a moves with its extremities in 
two rectangular axes ; find the envelope. 

13. Show that the envelope of 

Acr + Ba + C= o, 
where A, B, and C are functions of x and j>, is 

4AC- B 2 = o. 

14. Find the curve to which the lines given by the equation 

y = mx ± \\arn 1 + bm 4- c) 
are tangent. 4 (ay- + fov + cx~) = ^ac — & Q . 

15. Show that the envelope of 

Aa s + Ba' + Ca + Z> = o, 
^4, .Z?, C, and Z> being functions of x and y, is 

18ABCD - 2 7 A 2 D* + ^C 2 - 4 (^fC 3 +B Z D) = o. 

16. The centre of a circle which passes through the origin moves 

on the equilateral hyperbola 

222 
x ■— y — # ; 

find the envelope. 

The lemniscata, (x 2 -\- y*) = 4# 2 (# 2 — y 3 ). 

17. Find the envelope of the parabolas which touch the coordinate 
axes at the distances a and fi from the origin, when afi = f, the equa- 
tion of the parabola being 

x~ l y 1 

• — ■ 4- — =3 1. 

a* pi 1 6xy = <r 2 . 



400 APPLICATIONS TO PLANE CURVES. [Ex. XXXVI. 

1 8. The intercepts, a and /?, of a straight line on the coordinate 
axes are connected by the linear relation 

na + fi — c ; 
find the envelope. 

The parabola, (y — nxf — 2ncx — 2cy + c* = o. 

19; Find the envelope of the system of curves, 
when a m + fi m — <; w . 

m» »m ran 

20. By means of the result obtained in example 19, show that the 
envelope of a straight line, the sum of whose intercepts is constant, 
is a parabola touching the coordinate axes. 

21. From a point in the ellipse perpendiculars are drawn to the 
axes ; find the envelope of the line joining the feet of these perpen- 
diculars. 



2Y+feV«>. 



XXXVII. 

Envelopes of Systems of Straight Lines. 

378. In the case of envelopes of systems of straight lines, 
expressions for the coordinates in terms of the arbitrary para- 
meter can always be found, and, although the elimination of this 
parameter may be impracticable, the curve can always be traced ; 
as the following example will serve to show. 



§ XXXYlI.j ENVELOPES OF STRAIGHT LI. YES. 4OI 

A line AB of fixed length a slides be tic ecu rectangular axes : it 
is required to determine the envelope of a perpendicular to this line 

passing through the extremity A that slides on the axis of x. 

Denoting by a the angle between AB and the axis of x, we 
have for the intercept OA the expression a cos a, and for the 
equation of the perpendicular 

y — cot a (x — a cos a), 

or y tan a — x — a cos a — o (i) 

Taking the derivative with reference to a, we have 

y sec 2 a — a sin a = o, 

or y = a sin « cos 2 a ; (2) 

whence, substituting in (1), we derive 

x = <? sin 2 « cos « + # cos <* (.3) 

Equations (2) and (3) give x and y in terms of a, and hence 
determine the curve. It is evident that the curve is symmetri- 
cal to both axes, since ± a and tc ± a determine values of x and 
y differing only in sign. 

By differentiation, we obtain 

-J- — CI cos a (cos 2 a — 2 sin 2 a), .... (4) 
da 

and — = a sin a (cos 2 a — 2 sin 2 a) (5) 

da 

Corresponding to a = o, we have the point (a, o) at which the 
curve cuts the axis of x at right angles. As a increases from 
zero, equations (4) and (5) show that x and y increase until 

cos 2 a — 2 sin 2 a = o or tan a — J ^2, 
and that they subsequently decrease simultaneously; the point 



402 APPLICATIONS TO PLANE CURVES. [Art. 378. 

corresponding to this value of a is therefore a cusp. The arc 
generated as a passes from this value to 90 , touches the axis 
of x at the origin. 



The Evolute regarded as an Envelope. 

379. Since the evolute of a given curve is the curve to 
which all the normals to the given curve are tangent, it is 
evidently the envelope of these normals. 

The equation of the normal at the point (x, y) of a given 
curve may be written in the form 

x'-x + (y'-y)£ = o, (1) 

(x',y) being any point of the normal. 

dy 
In this equation y and -~- are functions of x determined by 
ax 

the equation of the given curve, and x is to be regarded as the 

arbitrary parameter. Hence, differentiating with reference to 

x, we have 

-'-©Vc/^g- ( 2 ) 

Equations (1) and (2) are equivalent to the equations given 
in Art. 354. Hence it is only when the equation of the normal 
is expressed in terms of some other parameter that this method 
differs essentially from that previously given. 

380. To illustrate, let us determine the evolute of the ellipse, 
employing the equation of the normal in terms of the eccentric 
angle. 

The equations of the ellipse are 

x— a cos tpj and y — b sin ip ; 

whence dx — — a sin tpdip, and dy — b cos t/> dip. 



§ XXXVII.] THEEVOLUTE REGARDED AS AX ENVELOPE. 403 

Substituti n in the equation of the normal, 

— x) dx - (y — y)dy — o, 

gives ax' sin </• — by' cos >/■ — \cf — b~) sin //• cos ip — o. 

Differentiating, we have 

ax' cos -/• — by sin //• — (a'- — b' : ) (cos* //• — sin s //■ 1 = o ; 
eliminating ;■' and x' successively, and dropping the accents, 
ax — (a* — b' : ) cos 3 tp and by = — u?" — b r ) sin 3 ?/■ ; 

whence { ax ) :i + (jfyfi = (** — ^ r 



Negative Pedals. 

381. In Fig. 78 P, the foot of the perpendicular from the 
origin upon the tangent PP' to a curve, de- 
scribes the pedal of this curve ; hence the | A 
curve to which PP' is tangent is the negative 
pedal (Art. 308 1 of the curve described by P. ^^ 
Hence the negative pedal of a given curve is the \^ / x 
envelope of the perpendicular drawn through the ''^v/p 
extremity of the radius vector from the pedal 
origin. 

If 6, the vectorial angle of the point P on the given curve, 
be taken as the arbitrary parameter the equation of the per- 
pendicular will be 

x cos $ - y sin 6 — r, (1) 

dr 

whence — x sin 6 + y cos 6 = — ; (2) 

ciu 




404 APPLICATIONS TO PLANE CURVES. [Art. 38: 

and, eliminating j and x successively, we obtain 



x — r cos 6 — — sin 
do 



dr 
y — r sin + — cos 6* 



(3) 



the rectangular coordinates of the negative pedal. 

382. For example, let it be required to determine the negative 
pedal of the strophoid, the node being the pedal origin. 

The polar equation of the strophoid referred to its node is 
[see equation (4), Art. 267] 

a cos 26 , n . .. fl ; m 

r = _ = a (cos V — sin tan 0) ; 

cos J 

dr 
whence — = — a sin 6 (2 + sec 2 6). 

au 

Substituting in equations (3) of the preceding article and 
reducing, we have 

x — a sec 2 (9, and y = — 2a tan 6 ; 

whence, eliminating 6, 

j 2 = 4a (x — a). 

Hence the negative pedal is a parabola whose vertex is situated 
at the point (a, o), the vertex of the strophoid. 



* Since it is shown in Art. 350 that the values of ip are identical at correspond- 
ing points of a curve and its pedal, the value of PP' is r cot ip ; hence these ex- 
pressions for x andjy could have been derived by projecting the broken line OPP' 
on the coordinate axes. 



§ XXXVII.] 



RECIPROCAL CURVES. 



405 



Reciprocal Curves, 

383. The reciprocal of a given curve is defined in Art. 309 
as the locus of the pole of the tangent to the given curve ; that 
is, PR, Fig. 79, being tangent to the given curve, the reciprocal 
is the locus of P when 

OR- OF = k\ 

The locus of R is the pedal (see Art. 307) of the given 
curve, while the locus of P is the inverse (see Art. 306) of that 
of R. Now it was shown in Art. 350 that the values of tp (the 
inclination of the curve to its radius vector) are identical at P 
and R, these being corresponding points of a curve and its 
pedal. Again, since the radius vector of the inverse of a given 
curve is k 2 u, it is obvious, on comparing equation (3) of Art. 318 
with equation (6) of Art. 319, that the values of >/: at R and P 
are supplementary ; hence the values of \p at P and P are 
supplementary. 

384. We shall now show that the given curve is the reciprocal 
of its own reciprocal. In other words, it is to 
be proved that P is the pole of the tangent to 
the reciprocal curve at P. 

Let this tangent meet OP in R' . The angles 
OPR and OP R' are equal by the preceding 
article ; hence the angle at R' is a right angle, 
and the similar triangles give 

OP- OR' = OR- OF = k 2 

Therefore P is the pole of P'R, and the locus of P is the reci- 
procal of the locus of P. 

It follows that the reciprocal may be defined as the envelope of 
the polar of a point on t lie given curve, or as the negative pedal of 
the inverse of the given curve. 




406 APPLICATIONS TO PLANE CURVES. [Art. 385. 



385. If r and 6' denote the polar coordinates of P', we 
have 

x — r cos 6', y = r ' sin 0\ OR = — ; 

and the rectangular coordinates of R are 

k" cosd' , k'smd' 
; and * ; . 



The equation of PR, since it passes through the point R 
and is inclined to the axis of x at the angle 6 + \n, is 

/P sin ff a , f P cos & 

y — — — ; — ■ = — cot I x , — 



or r'y sin 6' — k? sin 2 & + r'x cos & — fc* cos 2 6' = o; 

that is xx' + yy — k\ (1) 

which is therefore the equation of the polar of P'. 

The reciprocal may now be regarded either as the locus of 
P\ when the line (1) is tangent to the given curve ; or as the 
envelope of the line (1), when P' is a point of the given curve. 
When the latter view is adopted x and y must be regarded as 
two parameters connected by the equation of the given curve, 
and the envelope is found as in Art. 376. See Examples 14 
and 15, below. 

386. The reciprocal of a conic zvith reference to any point is a 
conic. 

Adopting the former of the two methods suggested in the 
last paragraph of the preceding article, we take the given point 
as an origin and denote by P z a point of the given conic. The 
equation of the given conic may be assumed in the homoge- 
neous form 

Ax\ + 2Bxiy t + Cyl+ 2Dax x + 2Eay 1 + Fa 2 = 0. 



§ XXXVII.] 



RECIPROCAL CURVES. 407 



The equation of the tangent at l\ is, by equation (4), Art. 313, 

x(Ax T +/>>! +Da)+y{Bxi + £> x + Ea) + a(Z)x l +Ey* + Ea)=o. (1) 

Denoting the pole of this tangent by P 31 and putting — ab in 
place of /"' in formula (1) of Art. 385, its equation may also be 
written in the form 

XX* + yy 2 + ab — o (2) 

Now, since (1) and (2) are equations of the same line, the 
coefficients of x, y, and a in these equations must have a com- 
mon ratio, and, denoting this ratio by /, we may therefore 
write 

Ax, + By, + Da— lx 2 



Bxi + Cy* + Ea = fy 9i y (3) 

* Dx x + Ey, + Fa = lb. J 

These equations when solved for x lt y lf and a will obviously 
give expressions for these quantities of the first degree with 
respect to x 2 , y 2 , and b. 

Now, since x x y z is on the line (2), we have 

x x x 9 + y x y 2 + ab = o, 

and, if in this equation we substitute the linear expressions for 
x lt y Jf and a, the result will obviously be of the second degree 
with respect to x 2 , y 2 , and b ; hence the reciprocal curve is a 
conic* 

* If we denote by A the symmetrical determinant of equations (3), and by a, fi, 
etc., the minors corresponding to A, B, etc., the solution of these simultaneous 
equations is 

Ax, = / («x, + 0y % + Sb), 

Ay, = I (J3x 2 + yy 2 + eb), 

Aa = I (8x* + F.y 2 + <pb) ; 
and the substitution gives 

exxj + 2(ix, 2 y. 2 + yy<? + 2dbx 2 + 2£by 2 + <pb- = o. 



408 APPLICATIONS TO PLANE CURVES. [Art. 387. 



Caustics. 

387. When a system of rays of light in a plane is reflected 
or refracted at a given curve, the reflected or refracted rays 
will in general form, not a pencil of rays passing through a com- 
mon point, but a system having an envelope. This envelope 
is called a caustic of the given curve ; the point from which the 
rays emanate is called the radiant point. 

When the caustic has a cusp, a large number of the rays 
pass nearly through this point ; hence the cusp of a caustic is 
called the focus (burning point) corresponding to the given posi- 
tion of the radiant point. 

388. To find the caustic by reflection of a circle when the ra- 
diant point is on the circumference. 

Let AP represent an incident and PQ a 
reflected ray. Denoting the angle PA by 
0, we have, evidently 




Fig. 80. 



APQ=2d, and PRX = 3d. 
The coordinates of P are 
a cos 26 and # sin 20; 
hence the equation of PQ is 

y — a sin 20 = tan 3$ (x — a cos 2(9), 

or y cos 3$ — x sin $6 — — a (sin 3<9 cos 20 — cos 36 sin 20) 

= — a sin 6 (1) 

Taking derivatives, we deduce 

y sin $6 + x cos $0 = \a cos & ; .... (2) 
eliminating y from (1) and (2), we express x in terms of 0; thus 
x — \a cos 3# cos 6 + a sin 3 (9 sin 6. 



§ XXXVII.] CAUSTICS. 409 

This equation may be simplified in the following manner: 

x = (fa — %a) cos 3# cos 6 + (fa + ^z ) sin 3^ sin ; 

whence x = frt cos 2^ — Ja cos 4^ (3) 

In like manner, we find 

^ = $# sin 26 — \a sin 4O (4) 

Equations (3) and (4) show that the caustic is an epicycloid ; 
the radius of the rolling circle, and also that of the fixed circle, 
being \a : but it is shown in Art. 296 that in this case the epi- 
cycloid becomes a cardioid. 

Examples XXXVII. 

1. From any point C on the circumference of a circle whose radius 
is a an ordinate to the fixed diameter AB is drawn, and through the 
foot of the ordinate a perpendicular to the chord AC is drawn ; find 
the equations of the envelope of this perpendicular, and trace the 
curve. 

Denoting by a the angle BAC, and taking the origin at A, we 
derive 

x = 2a cos 2 a (3 — 2 cos 2 a) ; 
y = — 4a sin a cos 3 a. 

The curve is symmetrical to the axis of x. a = o gives the point 
(20, o), a = 30 gives a cusp, and a = 90 a cusp at the origin. 

2. From any point B on the circumference of a circle whose radius 
is a an ordinate is drawn to the fixed diameter AC, and the foot of the 
ordinate is joined to the middle point of the chord AB ; find the en- 
velope of the joining line. 

Denoting the angle BAC by 0, and taking the origin at A, we 
derive 

x — 2a cos 2 cos 2fj : 
v = 2a sin 2 sin 20. 



4IO APPLICATIONS TO PLANE CURVES. [Ex. XXXVII. 

This curve, like the curve determined in the preceding example, is 
the three-cusped epicyloid, the origin being in this case a vertex. 

3. The points A and B move uniformly on the circumference of 
the circle whose radius is a, the ratio of the rates of these points 
being m : n; find the envelope of the chord AB. 

Denoting by mtp and wp the inclinations of the radii OA and OB, 
respectively, to the axis of x, the equation of the chord is 

x cos -J (m -f n) ip + y sin ^ {in -f n) ^- a cos \ (m — n) tp ; 

whence we derive the equations of the envelope ; viz., 

am , an , 

x — cos mp H cos mw> 

m + n m + n 

am an 
and y = sin nip + ■ sin mip. 

y m + n m + n 

This envelope is an epicycloid or a hypocycloid according as m and 

n have the same or opposite signs. The vertices are in both cases 

on the given circle, and the cusps are on the circle whose radius is 

m — 11 

a. 

111 + 11 

4. Find the envelope of a line which revolves uniformly about a 
point which moves uniformly in a fixed straight line. 

Taking the fixed line as the axis of y, and, for the origin, the posi- 
tion which the moving point occupies when the revolving line coincides 
with the fixed line, the equation of the revolving line is 

x = tan a (y — aa); 

whence x — a sin 2 a and y = a sin a cos a -f- aa, 

or introducing in the double angle 

a , N a , . v 

x = - (1 — cos 2a), y — - (2a + sin 2a). 

The curve is therefore a cycloid referred to its vertex. (See Art. 290.) 



§ XXXVII.] EXAKP1 411 



5. Find the equation of the evolute of the parabola, using the equa- 
tion of the normal in terms of its direction-ratio ; viz., 

y = mx — zdm — am*. 

2~tn' = 4 (x — 2,1 ) ". 

6. Find the equation of the evolute of the cycloid by means of the 
equation of the normal in terms of //•. 

The equation of the normal is 

sin ih , 

x + v r — aw = o. 

"1 — cos ip 

The equations of the evolute are 

x = a ($ + sin ?/•) and y = — a (1 — cos ?/-). 

(Compare Art. 357.) 

7. Show that the equation of the normal to the curve 

x = a cos a (1 -f sin 2 a), v = ^ sin a cos 2 or 

is ^ sin a -f- r cos a = a sin 2 a ; 

and thence prove that the evolute is the four-cusped hypocycloid. 

8. Find the equation of the evolute of the hypocycloid. Art. 297. 
The equation of the normal is 

a — 2b , . a — 2b . a . 

x cos — — : — ib — y sin = w = a cos — r w ; 

2b 2b 2b 

( a — b b a — b \ 

whence x = a i f- cos ip cos — = — ip r . 

( a — 2b a — 2b b ) 

a— b . , b . a — b . ) 
sin ip H 7- sm —7 — ip f . 



a — 2b a — 2b 



412 APPLICATIONS TO PLANE CURVES. [Ex. XXXVII. 

9. Determine the negative pedal (see Art. 381) of the parabola 

y 1 = 4ax. 
The semicubical parabola (x — 4a) 3 = 2 jay* 

10. Determine the negative pedal of the cissoid 

sin 2 
r = 2a 



cos 

y* -— — 8ax. 

11. Prove that the negative pedal of the spiral of Archimedes is the 
involute of the circle. 

12. Determine the negative pedal of the curve 

r = b sin mQ. 

x = b sin mQ cos — tnb cos mQ sin : 
y = b sin mQ sin + mb cos mQ cos 0. 

By putting for 0, # 0, and for mo. H 0, 

J r 2 2 2 2 

it may be shown that this curve is a hypocycloid. 

13. Derive the polar equation of the negative pedal of the curve 

r m = a m cos mQ. 
The rectangular equations are 

1 - m 1-to 

# = a (cos ^0) m cos(i —m) Q, y = a(cosmQ) m sin(i — m)Q. 
Whence, denoting the polar coordinates of the pedal by r and 0', 

■ y . — 

tan Q'= — — tan (1 — m) 0, and /— # (cos ^0) ra • 

x 

therefore, eliminating 0, 

» ^_ ^0' 

r ^^ = tf 1_OT cos 



1 — m 



§ XXXVII.] EXAMPLES. 413 

14. Find the reciprocal of the parabola referred to its vertex. 
Equation {1), Art. 5S5, is 

xx'+y/ = i\ 

and treating the reciprocal as an envelope, the relation between the 
parameters is in this case 

= 4ax'. 

Hence the reciprocal curve is ay* = — Zfx. 

15. Determine the reciprocal of the circle 

The result is a" {x* + /) = (# - bx)\ 

the equation of a conic referred to its focus and axis. This conic is 
an ellipse, an hyperbola, or a parabola, according as the origin is 
within, without, or on the circumference of the circle. 

16. Find the caustic by reflection for the cycloid, the incident rays 
being perpendicular to the base. 

The inclination of the reflected ray is \tt — ?/-, and its equation is 

y sin ip — x cos ip = a (sin ip — ip cos tf>) ; 
whence x — a (t/> — sin tf> cos ip), and y = a sin 2 ip : 
or, introducing the double angle, 

x = \a (2>l- — sin 2^) and y = \a (1 — cos 2ip). 
The caustic is therefore a cycloid. 

17. Find the caustic by reflection for parallel rays, the curve being 
a circle. 

Putting the origin at the centre, and the axis of x parallel to the 



4H APPLICATIONS TO PLANE CURVES. [Ex. XXXVII. 

incident rays, the coordinates of the point of incidence are a cos 6 and 
a sin 0, and the equation of the reflected ray is 

y cot 20 — x + \a sec 6 = 0; 

whence y = a sin 3 0, and x = \a cos (3 — 2 cos 2 0). 

These are the equations of a two-cusped epicycloid, the radius of the 
fixed circle being \a, and that of the rolling circle \a. 

18. Find the caustic by reflection of the parabola, the incident rays 
being perpendicular to the axis of the curve. 

Taking the origin at the focus, the equation of the parabola is 

y 2 = 4# {x + a); 

and, denoting by (f> the inclination of the normal to the negative direc- 
tion of the axis of x, we have 

y — 2a tan ^ and x — a (tan 2 6 — 1). 

The equation of the reflected ray is 

y — y = tan (f n — 2(f)) (x — x') — cot 2$ (x — x'), 

which reduces to 

y sin 2 (f> — x cos 2 (/) = a sec 2 ^. 

^ = — a sec 3 <^cos 3^ and jy = sec 3 <^sin3^ ; 

therefore the polar equation of the caustic is 

?-3 = a z sec ^jt — q 



CHAPTER XI. 
Functions of Two or More Variables. 



XXXVIII. 

The Derivative Regarded as the Limit of a Ratio. 

389. The difference between two values of a variable is fre- 
quently expressed by prefixing the symbol A to the symbol 
denoting the variable, and the difference between correspond- 
ing values of any function of the variable, by prefixing A to the 
symbol denoting the function. Hence x and x + Ax denote 
two values of the independent variable, and A fix) denotes the 
difference between the corresponding values of f{x) ; that is, 
My = fix), 

Ay = Jf(x)=f{x+Jx)-f(x). ... (i) 

If we put Ax = o, we shall have Ay = o ; 

hence the ratio ~ — J -± - 1 J v ' (2) 

Ax Ax } 

takes the indeterminate form - when Ax = o. The value as- 

o 

sumed in this case is called the limiting value of the ratio of the 
increments, Ay and Ax, when the absolute values of these incre- 
ments are diminished indefinitely. 

390. To determine this limiting value, for a particular value 
a of x, we put a for x and z for Ax in the second member of 



41 6 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 39O. 

equation (2), and evaluate for z = O, by the ordinary process 
(see Art. 96). Thus 

f(a + z)-f(aj 



'/'(«) (3) 

Therefore when zl;r is diminished indefinitely, the limiting value 

, and, since a denotes any 
value of X, we have in general 



of 1 -~- corresponding- to x = a is —f- 
Ax ^ * dx_ 



limit of -f- — -j-. 
J Ax dx 

A y 
If we denote by e the difference between the values of —- and 

J Ax 

—-. we shall have 
dx 

%=%+- <4) 

and the result established in the preceding article may be ex- 
pressed thus — 

e = o when Ax = o ; 

in other words, e is a quantity that vanishes with Ax, 



Partial Derivatives, 
391. Let u = f(x,y), 

in which x and y are two independent variables. The deriva- 
tive of u with reference to x, y being regarded as constant, is 

denoted by ■=- u, and the derivative of u with reference to y, x 
dx 

being constant, by -=- u. These derivatives are called the pan 
tial derivatives of u with reference to x and y respectively. 



§ XXXVIIL] PARTIAL DERIVATIVES, 417 

Adopting this notation, the result established in Art. 64 may 

be expressed thus : 

du = — — u ' dx 4- -z—U'dy; 
ax ay 

provided u denotes a function that can be expressed by means of the 
elementary functions differentiated in Chapters II and III. 
It is now to be proved that this result is universally true. 

392. Let A x u denote the increment of u corresponding to 
Ax, y being unchanged, A y u the increment corresponding to Ay, 
x being unchanged, and Au the increment which u receives 
when x and y receive the simultaneous increments Ax and Ay. 
Let 





u' -f{x + Ax,y), 




and 


u" —fix + Ax, y + Ay) ; 




then 


A.rU = U' — U, 

Ayit = u" — u\ 




and 


Au = u" — u ; 




hence 


Au = A x u 4- A y u\ 


• • (1) 



Denoting by At the interval of time in which x, y, and u re- 
ceive the increments Ax, Ay, and Au, we have 

Au A x u A v ?i f f N 

— = — 1 — (2) 

At At At w 

Since Au is the actual increment of u in the interval At, the 

du 
limit of the first member of equation (2) is, by Art. 390,-^-, the 

A 11 
rate of u. The limit of — 4— is the rate which u would have 

At 



41 8 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 392. 

were x the only variable ; and, since -j— u • dx is the value which 

ax 

du assumes when this supposition is made, if we put 

-j-tc- dx = d x u, 
ax 

this rate will be denoted by -^— . Hence by equation (4), Art. 

at 

390, equation (2) becomes 

du , d x u . d v u' „ 

in which e, e' , and e" vanish with At ; but when At — o, Ax — o, 
and therefore u' = u: hence, putting At = o, we have 





du d x u _ d y u 
W ~ ~dt ^ ~dt' 


Therefore 


du = d x u + d y u\' 


that is, 


du — —i-u • dx + -;-u - dy. 
dx dy 



393. This result is usually written in the form 

7 du 7 du , 

du = -r~ dx -f -=— dy, 
dx dy 

but when written in this form it must be remembered that the 
fractions in the second member represent partial derivatives, 
the symbol du in the numerators standing for the .quantities 
denoted above by d x u and d y u, which are sometimes called par- 
tial differentials. The du that appears in the first member is 
called the total differential of u when.*- and y are both variable. 
For example, if u is an implicit function of x and y satisfy- 
ing- the relation 

u — o, (1) 



't> 



§ XXXVIII.] PARTIAL DERIVATIVES. 4*9 

dy 

and it is required to express ; in terms of the derivatives of 

ax 

u ; we have, by differentiating equation (i), 

du j dit . 

dx + — dy = o ; 
ax dy 

du 

i, dy - (i\ 

whence dx~~ — (2) 

dy 

This equation seems to be inconsistent with the principles of 
algebra ; but when we remember that the du in the numera- 
tor is in reality d x ?i, while that in the denominator is d y u t the 
difficulty vanishes, since we have from equation (i) 

du = d x u + d y u = o ; or d x u = — d y ti. 

394. The result deduced in Art. 392 is readily extended to 
the case of more than two independent variables. 

Thus if u —f{x,y, z t • ■ .), 

u' y u", etc. being defined as in Art. 392, it may be shown that 

An — A x ii + A y u' + A z u" + • • • , 
and thence that 

7 d j d r d , 

du = —j- u • dx + -=- u • dy + -=- u • dz + . . . 
dx dy dz 



The Approximate Vahics of Errors due to Small 
Errors of Observation. 

395. The principle embodied in equation (4), Art. 390, is 
useful in determining approximately the relation between the 



420 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 395. 

errors in any observed quantities, and the resulting error in a 
quantity derived from them by computation. 

To illustrate, let the data be the two sides a and b and the 
included angle C of a plane triangle, and let it be required to 
determine the third side c. 

The relation between c and the given parts is 

c 2 = a 2 + b 2 — 2ab cos C, (1) 

and it is required to ascertain the effect of certain small errors 
in the values of a, b, and C upon the resulting value of c. 

Let Aa denote the error in the value of a, and Ac the cor- 
responding change in the value of c ; that is, the change pro- 
duced by changing # to# -\-A a while b and C remain unchanged. 
Now, c being regarded as a function of a, we have, by Art. 390, 

Ac dc 

■ — = f- e. 

Aa da 

By differentiating equation (1), a and c being regarded as the 
only variables, we have 

c dc = (a — b cos C) da. 

Hence approximately, when Aa is small, 

Ac __ a — b cos C 
Aa ~ c 

or, since a = c cos B + b cos C, 

Ac = cos B -Aa. 

396. To indicate that this expression is the approximate 
value of the error due to the error in a, we use the notation 
adopted in Art. 392 : thus 

A a c — cos B • Aa (1) 



§ XXXVIII.] VARIATIONS OF TRIANGLES. 4 21 



In a similar manner we obtain 

Abe — cos A J b, and A ,,c = a sin BAC. 



If we apply successively the increments Aa, Ab, and AC, 
c will receive three increments; of these the second and the 
third will, when Ab and A C are small, differ very little from A/,c 
and z/ c c respectively. Hence the total error will differ very 
little from the sum of the partial errors whose approximate 
values are given above ; therefore we have, approximately, 

Ac = cos BAa + cos A Ab + a sin BAC. .... (2) 

It is obvious that this result may be obtained by substituting 
the symbol A for d in the expression for the total differential 
of c regarded as a function of a> b, and C; viz., 

dc — cos B da + cos A db + a sin B d C. 

In applying equation (2), it must be remembered that AC is 
expressed in circular measure. 

397. It should be remarked that the expressions for par- 
tial derivatives which involve the parts of a triangle depend 
not only upon the parts whose differentials are compared, but 
also upon the other two parts which appear among the given 
quantities. Thus, in the problem solved in the preceding 
article, 

dc . 

db= cosA > 

the other parts involved being a and C. 

But, if c is computed from the data b, A, and C, we have 

c sin (A + C) — b sin C; 

dc sin C 



whence 



db sin {A + Cy 



422 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 397. 

The dc in the numerator of each of these expressions might 
be denoted by d b c ; the difference in value is dependent on the 
fact, that in the former case the total differential dc is assumed 
to be decomposed thus— 

dc — d b c + d a c + d c c ; 

while in the latter case we assume 

dc == d b c + d A c + d c c, 

the partial differentials having radically different meanings in 
the two cases. 

Examples XXXVIII. 

i. Given u = (# 2 + y 2 )*, prove that 
du du a 

x + y — 2 Um 

dx dy 



XV 

2. Given u = — - — , prove that 

x + y 

du du 
dx dy 

3. Given u= tan -1 I — 3^/? prove that 



du , du 

x -j- +y — = o. 
ax dy 

^- , /- 1 du . du 

4. Given u = log^, to find — and — . 



du _ 1 du _ — logx 

dx x logjy ' dy y (logj') 2 ' 



§ XXXVIII.] EXAMPLES. 423 



5. Given u = log [x + V(x* +J'*)]> prove that 
' d d\ 



dy 

6. Given u — log (.v 3 +j' 3 + z 2 — 3^5), prove that 

</# */# //# -* 

1 1 — 2 1 

dx dy dz x -\-y + z 

7. In a plane triangle, determine the approximate value of AC 
when the data are the three sides. 

_ Ac — cos BAa — cos A Ab 
bsinA 

8. Find the approximate value of Ac when the data are the base 
line b and the adjacent angles A and C. 

Ac— a cosec BAC — c cot BAA + C ——. 

9. Find the approximate value oi AC when the four parts in- 
volved are A, C, a, and c. 

.-, sin A . tan C . . tan C A 

AC= ~ Ac + T AA Aa. 

a cos C tan A a 

10. In a plane triangle, given A, B, and c, determine/ (the perpen- 
dicular on c), and find the partial derivative of/ with reference to A. 






sin A sin B . dp c sin 2 B 

P ~ c ~ — Hi o\ » whence -~ — 



sin (^ + ^)' ^4 sin 2 (^ + ^)' 

11. In a plane triangle, given A, B, and/, find the partial deriva- 
tive of c with reference to A. 

dc cos (A + B) 

-T~a —P ~ — a- — 7T — c cot A. 

dA r sin A sin B 



424 FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXXVIII. 

12. The area k of a plane triangle being determined from two sides 
and the included angle, prove that 

Ak = i(b sin CAa + a sin CAb + ab cos CJC). 

13. The area k of a plane triangle being determined from the three 
sides^ prove that 

Ak rj „ „ „-. a a cot A 

= [b* +c* - a'] — = . 

Aa L J 8£ 2 

14. In a right spherical triangle, given sin a = sin A sin c, derive 

cot a Aa = cot AAA + cot cAc. 
Take logarithniic derivatives. 

15. In a right spherical triangle, given cos^ = cot A cot B, derive 

Ac AA AB 

+ 



2 cot c sin 2 A sin 2i? ' 

16. Determine the relation between the errors when the parts in- 
volved are the three sides and one angle of a spherical triangle. 

Ac— QosBAa + cos A Ab + sin b sin A AC. 



XXXIX. 

The Second and Higher Derivatives regarded as Limits. 

398. In Art. 390 it is shown that 
Ay dy 



+ e. 
Ax ax 



In this equation e is a function of x and likewise of Ax ; hence 
the derivative - T - is in general a function of ^ and of Ax, It is 



§ XXXIX.] THE SECOND DERIVATIVE ASA LIMIT. 4 2 5 



also proved in the same article that e becomes zero when Jx 
vanishes; that is, e assumes a constant value independent of the 
value of x when Jx becomes .zero ; hence, when Jx is zero, the 
derivative of e with reference to x must take the value zero, 
whatever be the value of x ; in other words, 

-=- vanishes with Ax. 
ax 

In a similar manner it may be shown that each of the higher 
derivatives of e with reference to x vanishes when Ax — o. 

399. Since — is a function of x, A — will denote the incre- 

Ax Ax 

ment of this function corresponding to Ax. Employing the 

symbol — — to denote the operation of taking this increment, 

and dividing the result by Ax, we obtain, by applying to this 
function the principle expressed in equation (4), Art. 390, 

A Ay d Ay , N 

- — • — • J 4- e . . . . . (1 ) 

Ax Ax~ dx Ax • ' W 

dx \dx 

= ^1 + *L + e '. 
dx 1 dx 

de 
In this equation both e' and -=- vanish with Ax by the preced- 
ing article ; hence the sum of these quantities likewise vanishes 
with Ax, and may be denoted by e. Thus we write 

A Ay dy , , . 

. , . —Z- = — rL -4- e (2) 

Ax Ax dx* W 



426 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 4OO. 

400. Since Ax is an arbitrary quantity it may be regarded 

Ay 
as constant, whence A — — is the increment of a fraction whose 
Ax 

denominator is constant ; but this is evidently equivalent to the 

result obtained by dividing the increment of the numerator by 

the denominator ; that is, 

A Ay = A-Ay ^ 
Ax Ax 

The numerator A • Ay is usually denoted by the symbol A 2 y ; 
hence equation (2) may be written thus : 



£=£+'. (3) 



Ax 2 dx 

and, since e vanishes with Ax, it follows that the second deriva- 
tive is the limit of the expression in the first member of equa- 
tion (3). 

In a similar manner it may be shown that each of the higher 
derivatives is the limit of the expression obtained by substi- 
tuting A for d in the symbol denoting the derivative. 

Higher Partial Derivatives. 

401. The partial derivatives of u with reference to x and y 
are themselves functions of x and y. Their partial derivatives, 
viz., 

d du d du d du . d du 

dx dx' dy dx' dx dy' dy dy' 

are called partial derivatives of u of the second order. 

It will now be shown that the second and third of these 
derivatives, although results of different operations, are in fact 
identical ; that is, that 

d du _ d du 
dy dx dx dy' 



§ XXXIX.] HIGHER PARTIAL DERIVATIVES. 427 

Employing the notation introduced in Art. 392, we have 

J_,, l =f{.v + A.v,j)~f(x,y); 

if in this equation we replace y by y + Ay, we obtain a new 
value of A x ii, and, denoting this value by d'u t we have 

A'u = f [x + Ax, y + Ay) - f (x, y + Ay). 

Since this change in the value of A x ti results from the increment 
received by y, the expression for the increment received by 
A x u will be A y (A x ?t) ; hence 

A y (A x u) = A' x u - A x u, 
or 

A y (A x u)=f(x + Ax, y + 4y)-f{x,y + Ay)-f{x + Ax,y)+f(x,y). 

The value of A x (A y ii) y obtained in a precisely similar manner, 
is identical with that just given ; hence 

A y {A x ii) = A x {A y ti) (I) 

Since Ax is constant, we have, as in Art. 400, 

4> ( A *u) _ A t 4r« 
Ax y Ax ' 

Hence, dividing both members of equation (1) by Ax • Ay, we 
have 

Ay A^u _ 4r AyU , , 

Ay' Ax Ax' Ay' K) 

or, employing the symbol — as in Art. 399, 

A A A A 

Ay Ax Ax Ay 



428 FUNCTIONS OF TWO OR MOKE VARIABLES. [Art. 4OI. 

From this result, by a course of reasoning similar to that em- 
ployed in Art. 399, we obtain 

d du _ d du . 

dy dx dx dy ' 

402. The partial derivatives of the second order are usually 
denoted by 

d 2 u d^u d*u 

dx* ' dx dy' dy- ' 

the factors dx and dy in the denominator of the second being, 
by virtue of formula (3), interchangeable, as in the case of an 
ordinary product. 

The numerators of the above fractions are of course not 
identical. Compare Art. 393. 

Formula (3) of the preceding article is readily verified in 
any particular case. Thus, given 



. du , . du „ T 

whence — = y x log y, and - T - —xy x ~ x \ 
dx dy 

d du , . N d du 

-7 y — y x ~ T {* logy + 1) = -j- - ~r- 

dy dx y v fe ^ J dx dy 

403. Equation (3) of Art. 40 1 shows that a differential ex- 
pression of the form 

Mdx + Ndy, 

in which M and N are functions of x and y, does not always 
form the total differential of a function of x and y. For if we 
assume the existence of a function u fulfilling the condition, 

du-Mdx + Ndy, (i) 



I XXXIX.] HIGHER PARTIAL DERIVATIVES. 4-9 

we shall have M- and X — . 

ax ay 

hence, by the equation cited above, we must have 

dM __ dX ( 

dy " dx ' - ' W 

Equation (2) constitutes therefore a necessary condition in 
order that the expression Mdx -f Ndx may be an exact differ- 
ential. 

404. The theorem proved in Art. 401 may be expressed 
thus — the operations of taking the derivative with respect to tzvo 
independent variables are commutative; that is, they may be 
interchanged without affecting the result obtained. 

This theorem may be extended to derivatives higher than 
the second, and also to functions of more than two independent 
variables. For it has been proved that we may, without affect- 
ing the result, interchange any two consecutive differentiations, 
and it is obvious that, bv successive interchanges of consecutive 
differentiations, we can alter the order of differentiation in any 
manner desired. Hence all differentiations with respect to in- 
dependent variables are commutative. 

In accordance with this theorem, the result of differentiating 
m times with respect to x, n times with respect to y, and p 
times with respect to z, may, without regard to the order of dif- 
ferentiation, be expressed by the symbol 

cT* n **n 



dx m dy n dz* 



Symbols of Operation. 

405. The symbol -j- has already been employed to denote 

the operation of taking the derivative, with respect to x, of 
the function to which it is prefixed, this derivative being a 



43° FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 405. 

partial derivative when the quantity is regarded as a function 
of more than one variable. So likewise the compound symbol 

— • -j- indicates the operation of taking the derivative with 

reference to x, and the derivative of the result with reference 
to y ; the symbol written last being applied first, since the com- 
pound symbol is prefixed to the function. It has however been 
shown in Art. 401 that in this case the operations are commu- 
tative ; that is, that 

d d _ d d 
dx dy dy dx' 

The compound symbol is called the symbolic product of the 
simple symbols. In this case the product is commutative like 
an ordinary algebraic product. 

The quantity affected by a symbol of operation is called 
the operand. When the symbolic notation is employed, it is to 
be understood that the product of all the factors following a 
symbol of operation constitutes the operand. 



Commutative and Distributive Operations, 
406. When m is constant we have 
d (mu) = mdu ; 
hence when u is a function of x 

d d r v 

-—-fnu — m—r-u (1) 

dx dx 

This equation indicates that the operation of multiplying by a 
constant, and the operation of taking the derivative with refer- 
ence to x, are commutative. It follows that the factors of the 



§ XXXIX.] COMMUTATIVE OPERATIONS. 43 l 

symbolic product y , when y is a variable independent of x } 

are commutative ; for, in performing the operation -r~,y is re- 
garded as a constant. 

On the other hand, x —— is not commutative ; for 
dx 

d du 

— XU = X — + u, 

ax ax 

, ., d du 

while x-j-u — x-y- . 

dx cix 

407. A repeated application of the same symbol, whether 
simple or compound, is indicated by affixing an index to the 
symbol : thus — 



«)' 



y dx" y dx~' 



and, since the operations indicated by the symbol are commu- 
tative, we have 



d\ 2 d* /N 

On the other hand, the operations indicated by the symbol x — 

(1 \ 2 /2 

x — - J is not equal to x" 1 — - . 

The result obtained by adding or subtracting the results 
arising from the application of two operative symbols may be 
expressed by connecting the symbols by the appropriate sign : 
thus — 

d \ du r . 

-— + m )« = -—+ mu (2) 

dx J dx 



43 2 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 407. 

Since the sum expressed in the second member of this equation 
is commutative, the symbolic sum expressed in the first mem- 
ber is likewise commutative. 

408. When u and v are functions of x we have 

d f . \ d d , N 

s (». +W )= s , + _ f ,. ...... (1) 

This equation signifies that the operation of taking a derivative, 
as applied to a sum, is a distributive operation ; that is, the re- 
sult which arises from performing this operation upon a sum 
is identical with the sum of the results obtained by performing 
the operation upon each quantity separately. Since this dis- 
tributive principle is applicable to ordinary algebraic multipli- 
cation, the first member of equation (1) is expanded exactly as 
it would be, if the symbols represented algebraic quantities. 
Equation (1) Art. 407 expresses the fact that the application 

of an exponent to a symbolic product of the form y -=- is dis- 
tributive ; and it is obvious from the mode in which this equa- 
tion arises that an exponent is distributive whenever the sym- 
bolic product to which it is applied is commtitative. 



Symbolic Transformations. 

409. The formulas of algebraic expansion are consequences 
of the commutative and distributive nature of algebraic multi- 
plication ; hence it follows that a symbolic product or power 
may be expanded by these formulas ; provided all the factors 
involved represent commutative operations. Thus — 



\dx J \dx J 



d \ / d \ d u 9 

u — -— - — au. 
dx* 



g XXXIX.] SYMBOLIC TRANSFORMATIONS. 433 



Again the total differential of a function of x and y is ex- 
pressed by the equation 

, , du . du [yd , d\ 

die = dx + dr — = dx — + dy-- u, 
ax y dy \ dx ' dv 



in which all the factors have commutative products, since dx is 
regarded as a constant in differentiating with reference to x as 
well as in differentiating with reference to y. Hence we have 



,, /, d , dy , d"u j . d"u 

d u— dx — + dy — ) u = dx" -7— + « dx"- 1 dy , r + • • , 

\ dx dy J dx" dx"- 1 dy 

a formula giving the nt/i total derivative of a function of two 
variables. 

410. The result deduced below is frequently employed in 
transforming operative symbols. 

Let u denote a function of 6 ; then we have, by differentia- 
tion, 

and multiplying by £~" e 

e -"°i e " Ku= (i + n ) u (I) 

Applying now the symbols whose equivalence is expressed in 
this equation to the equation itself, we have 

and, by repeating this process, we have in general 

e-**-£.0*..u=(4z + n) r 'U* .... (2) 
dff' \d0 J 



* This equation is equivalent to the result obtained by means of Leibnitz' 
theorem in Art. 90. 
28 



434 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 411. 



Eulers Theorem concerning Homogeneous Functions. 

411. A homogeneous algebraic function of the /zth degree 
involving two variables may be put in the form 

"=*"/£) « 

In this expression n admits of fractional and negative values, 
and moreover u may be a fraction whose numerator and de- 
nominator are homogeneous, n being in this case the difference 
between the numbers expressing their degrees. Thus — 

Vx + Vy 

u — —7 ~ 

x{x +y) 

is a homogeneous function in which n = — f . 

In equation (i)/may denote a transcendental function, and 
in this case u is still called a homogeneous function. It is to be 
noticed that when u is transcendental the expression under the 
functional sign /must be of the zero degree. Thus — 

u — log \x + v(x~ + y )] 

is not a homogeneous function because the quantity under the 
sign log, although homogeneous, is not of the zero degree. 

412. By differentiating equation (1), we derive 

£—-/©-*- •>/'©■ 



dy 

c 
dx ' dy 



. du , du m x ( y\ 

whence x—r- + y—- = nx n f ( - ) = nu. 

dx dy \xJ 



§ XXXIX.] EULER'S THEOREMS. 435 

or, symbolically, \x — + y-r)u = nu, (i) 

when u is a homogeneous function of the //th degree. 

Again, the derivatives of u are homogeneou ; functions of 
the (n — i)th degree ; hence, by the theorem expressed in 
equation (i), we have 

( d d\ du . s du . f d d\ du , s du 

K*Z +y dy) Z =( - n ~ l) Z' and Vdx +y Jy) Ty^^Tr 

whence, expanding, 

d 2 u d 2 u , x du , d'u d'u . x du 

X JS +y WZ = {n - l) di' and X dy-Z +y dy' =( "~ l) dy ; 



multiplying and adding, 

du t du 
Tx ' y ~d~y 



„d 2 u d 2 u ,d 2 u , N 

X 'd? +2Xy Jx-Ty+ y W ={n ~ l) 



. 2 d'u d u .d'u , . , , 

hence x~ — — 4- 2.fi/ -= — =- + jr -=-=- = n (n —i) u. . . (2) 

dkr y ax dy dy 

The results expressed in equations (i) and (2) and similar 
results involving higher derivatives are known as Eulers 
Theorems. 

Examples XXXIX. 

1. Given u = sec (y + ax) + tan (y — ax), prove that 

d 2 u _ 2 A 
dx" dy 2 ' 

tt -r 1 1 d 2 U d?U . • / 2\ 

2. Verify the theorem — — — = — — — when u = sin (xy ). 

3 dxdy dydx J ' 

TT ., . , d 2 u d 2 u . . / . a\ 

3. Verify the theorem — — — = — — — when u = log tan (ax + y ). 
dxdy dydx 



43^ FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXXIX. 



4. Verify the theorem 



5. Verify the theorem 



d 11 du . _* x 

7 o 7 = — — rr when u = tan — 

dy dx dx dy y 

d 3 u ofu 



dy dx 1 dx 1 dy 
6. Given u = sin.* cosy, prove that 

d i u d*u d*u 



when u = y log (1 4- xy). 



dy' 2 dx' 2 dx 2 dy 2 dxdy dxdy' 
7. Given u = x*z 4 + e x y 2 z z -f a^V, derive 



fl&f * ^ dfe 



6ytz 2 + 8y*. 



8. Given 



\f( A ab-c 2 ) 



, prove that 



dfc 2 da db 
9. Given & = (x + y) 2 , prove that 



d*u 



d 2 u du 



dx' 2 ' dxdy dx' 



10. Given u — log (V + _y 3 + z* — $xyz), prove that 

d 2 u d 2 u d 2 u d 2 u d 2 u d 2 u 

dx 2 dy 1 dz 2 dx dy 



dy dz dz dx 



(x + y + z) 2 ' 



Employ the symbol ( — 1 — - H — 7 ) u, and see Ex. XXXVIII, 6. 

\dx dy dz J 

11. When u = (x 2 + y*Y ■> verify the formula 



d' 2 u 



d 2 u 



d 2 u 



x -— 2 + 2xy—_ — — + y 2 —5 = o. See Art. 412. 



^ 



dx dy dy 



§ XXXIX.] EXAMPLES. 437 



12. When u — (x* +y) * verify the formula 



ax ax ay J dy 

13. Given u — — r prove that 

d^u (T'u (f'u 

14. Determine the value of 



du du x 1 — \ 

x— hV-Ti when // = tan -1 

dx J dy' ax 



Solution 



Since tan u is a homogeneous function of the first degree, we have, 
by Euler's theorem, 

du du 



. du du ax (.v 2 — r)' 2 

whence x — + y ~r = * , , ., — Sr=-. 

15. If u = sin p, & being a homogeneous function of the «th degree, 

. . . du du 

determine the value or x — S- y -y- . 

dx dy 

du du 

x — h v -r- = w cos £'. 

16. Given u = - £ ^ 2 , prove that 

/ </V d f dy u r ioaS 4tfV~l 



43 8 FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XXXIX. 

17. Given u— t ax+by , find the third total differential by the for- 
mula deduced in Art. 409. 

d s u = (a 3 dx* + $d > bdx t dy +- $ab 2 dxdy 2 + b*dy 3 )s ax + blf . 

18. Given u = (x 2 +y 2 ) ¥ , find the second total differential. 

d*u = (y 2 dx 2 — 2xy dx dy + x 2 dy 2 ) (x 2 + y 2 ) ~ 2 - 



19. Apply the symbol a Q - 1- 2a x -7 — f- s a z -7- to the expression 

22, 3,3 22 

<Zo #3 — Otfo #i #2 #3 + 4#0 #2 + 4#i #3 3^1 #2- 

Result o. 

20. Operate on a Q a 2 a 4 + 2a x a*a z — fl a| — 0? a* — a| with the symbol 
d d d d 



do- \- 2a z - 1- 3^2 -j h 4^3 

da T da* da$ da± 



Result o. 



21. Show that the symbols y—r- and a Q — 7 — f- 2a 1 — \- %a*—,— 

dx dai da* da 3 

give the same result when applied to the expression 

(a a* — &i) x 2 + {a Q a % — a* a*) xy + (a-LO* — a~*)y*. 

t> i.i_ , „ d z. d* jx y/x 

22. Prove that 8 — — x' 2 -r- 9 £ = £ . 

dx dx 

23. Show that the result obtained by applying the symbol 

r r' 

\dx) X \dx) t0 1.2 -.-r 1S 2-+ 1 - 1- 2.../ ' 

in which r' — r — 2n — 1. 

24. Show that the result obtained by applying the symbol given in 

^ x 
the preceding example to e^ x is ^ . 

Expa7idi7ig the function by the exponential theorem, the application of 
the symbol to any term of the series produces a preceding term. 



§ XL.] CHANGE OF THE INDEPENDENT VARIABLE. 439 

XL. 

Change of the Independent Variable. 

413. It is frequently desirable to transform expressions in- 
volving derivatives with reference to x into equivalent expres- 
sions, in which some variable connected with x by a known 
relation is the independent variable. This process is called 
changing the independent variable. 

Let/ denote the function whose derivatives occur in the 
given expression, and let 6 denote the new independent vari- 
able, the relation between x and 6 being, for example, 

;tr=tan# (i) 

To obtain an expression for the first derivative it is only 
necessary to substitute the value of dx derived from equation 
(i), viz., % 

dx = sec 2 6 dd ; whence -r-= cos 2 d~~ . . . (2) 

dx dd 

414. In the case of — 4- we cannot substitute the value of 
dx 1 

dx* ; because the d' 2 y in the numerator of this expression de- 
notes the value which this differential assumes when dx is con- 
stant, while the d*y in -=~ denotes the value assumed when dd 
au~ 

is constant (see Art. 80). We must therefore differentiate the 
expression for the first derivative and divide by the value of 
dx : thus from equation (2) we obtain 



dy d 



dx" sec 2 6 dd 



b*'%\ 



cos 2 # 



cos 2 6 -i — 2 cos 6 sin 6 -j- 
du~ du 



] • • (3) 



44° CHANGE OF THE INDEPENDENT VARIABLE. [Art. 414. 

In like manner an expression for the third derivative may- 
be obtained. 

415. As an application let it be required to transform the 
differential equation 

d*y 2x dy y _ 



dx* I + x 1 dx (1 + x'y 

into one in which 6 is the independent variable, the given rela- 
tion being x = tan 6. 

Eliminating^, -^- , and —-„ by means of equations (1), (2), 
dx dx 

and (3), we have 

, „ d 2 y * n ■ n dy 2 tan 6 „ n dy t y 

cos e ^k~ 2 cos 3 6> sin 0-^ + ^-cos 2 ^-^- + -^ = o; 

dv* dd sec 2 6 dv sec 

whence, reducing we have 9 

416. Expressions involving derivatives of 7 with reference to 
.r may be transformed into equivalent expressions involving 
derivatives of x with reference to y. In this case we have 

d*x dy d*x 

dy __ I m dy _ dy 2 dx dy 2 % 

dx~~~ dx .*'" d?~~ fd_x\ ~ " fdx\ 3 ! 

dy \dy J \dy) 

For example, by means of these substitutions the expression 

[-(!)7 . Mia 



p = 



becomes p = — 



75T ^ 2 



§ XL.] TRA NSFORMA TWN OF ( >/Y-//V. / 77 / r E S ) T MBi >/ S. 44 1 



Transformations of Certain Operative Symbols. 
417. If we put x — e e , we have 

Ox = e 9 d0 = xd0: 



whence 



d_ = d_ 

* dx ~ dB 



(0 



By differentiation we obtain 
d 



dx 



r.„ =X r-[ x *_ + 



whence by equation (i) 



''■••=[£]*'■"■ 



that is, the symbols ,r I_r -— x r and 

dx 






+ r 



(2) 



are equivalent. 



Now, putting r — I for r in these symbols, and applying 
them to equation (2), we have 



d 
x 12 -' —x r - x • x l 
ax 



x ' u = x 2 



dx- 



x r • It 



Lii +r ][i + r - l ] u > 



and, by repeated applications of this process, we have in gen- 
eral 



d* V d 

dx' 1 \_dd 



][i + r " x ] 



- d_ 

J6 



jL 



+ r — 2 



+ r — n + 1 



«■ • (>) 



In this equation r admits of negative and fractional values. 



44 2 CHANGE OF THE INDEPENDENT VARIABLE. [Art. 4 1 7. 



If we put r = o in equation (3), we have 



n d " _ d 

dx n dd 



' d 
dd 



— 1 



dd 



dd ~ n + l 



] u. . . (4) 



and if we put r — n in the same equation 



d? 

dx n 



d 



+ 7Z 



.dd 



+ ^ 



_</0 



+ 1 



u. . 



(5) 



4(8. Any two compound symbols of the forms occurring in 
the second members of the equations in the preceding article are 
commutative, since they involve differentiation and multiplica- 
tion by constants only ; therefore any two symbols of the forms 
given in the first members are likewise commutative. For ex- 

d m d p 

ample, the symbol -r-^ xm+p -7-7- admits of separation into two 

factors having the forms given in equations (5) and (4), respec- 
tively ; hence, commuting these factors, we have 



d m A . dp 



d m+p 

X& - ■ X 7 



dx m " dx* " dx m +£ 

the common value of these symbols being 



.dd 



+ m 



~d_ 

.dd 



+ m 



d_ 
dd 



] 



The essential characteristic of commutative factors of the 
form here considered is that the sum of the exponents equals in 
each case the sum of the indices of the derivatives. 



Expressions Involving Partial Derivatives. 

419. Let u denote a function of x and y, and let r and be 
two new independent variables connected with x and y by two 



g XL.] EXPRESSIONS INVOLVING PARTIAL DERIVATIVES. 443 

given equations. It is required to express the partial deriva- 
tives of u with reference to x and y in terms of derivatives with 
reference to r and 6. 

Now, since in this case u may be regarded as a function of 
r and 6, we have, by the theorem of Art. 392, 

, du . du 7n . . 

In accordance with the same theorem, if we suppose y to 
be constant du in equation (1) will become d x u which is the 

numerator of the partial derivative -r-. Upon the same sup- 

dx r 

position dr and dd become d x r and d x 6, while the ratios - and 

dr 

-jg, being independent of the absolute values of the differentials 

involved, are not affected by the supposition. Hence dividing 

by dx, we have 

d?i du dr du dd , . 
= I # (2) 

dx dr dx dd dx' K ' 

In like manner, making x constant, we derive 

du _ du dr du dd 
Jj~dr~d)i Jr de~dy (3) 

420. Let us now assume the given relations to be 

x = r cos V and y = r sin 6 (1) 

It is to be remembered that the four coefficients - r . -r- . — , 

dx dx dy 

in 

and — ■ are the partial derivatives of r and 6 with reference to x 
dy 

and y ; their values therefore are not to be obtained directly 

from equations (1), but from the expressions for r and 6 in 

terms of x and y. 



444 CHANGE OF THE INDEPENDENT VARIABLE. [Art. 420. 

Thus, from equations (1), we obtain 

r» = x 1 + y\ 
whence 

dr x 



-— = - = cos 6 

dx r 



dr y . 

— = - — sin 6 



dy r 



t nd 






6 = 


-- tan-i 

X 


dd _ 




' y 




sin 6 


dx 




r 2 




r 


dd 


X 




cos 


6 


dy 


r 2 




r 





• • • (3) 



The above method of proceeding should be carefully noticed, 

dx dx 
since the values of the partial derivatives —- , — - d , etc., which 

dr dv 

would be obtained by direct differentiation of equations (1) are 

not the reciprocals of the derivatives required ; for we obviously 

d r d> x 
cannot assume — — ■ -^ equal to unity.* 

Substituting in equations (2) and (3) of the preceding article 
the values of the coefficients given by equations (3), we have 

du du n du sin 6 , N 

J- x = ^ cose -M — - • • • • • (4) 

du du . n du cos /jN 

* The values of -r and -7- may also be found by elimination between the deri- 
vatives of the given equations with reference to x ; viz., 

dr dQ 

i = cos0^-rsin0-£, 

,dr ^aQ 

and O = sin G y + r cos -j . 

dx dx 

When, however, the given equations can be solved for the new variables, the process 
given in the text is preferable. 



§ XL.] EXPRESSIONS IN VOL VING PARTIAL DERIVA TIVES. 445 

421. Expressions for the higher derivatives may be ob- 
tained by differentiating equations (4) and (5). For example, 
let it be required to transform the expression 

d : u d'u 
dJ + ~df 

by making r and 6 the independent variables. 

In differentiating equation (4), it must be noticed that, since 

-J- is a function of r and 0. we have 
dr 

,du d fdu\ , d (du\ .„ 

d_du_ d'u dr d*u dd 
dx dr dr 2 dx drdd dx' 

The values of — and — are given in equations (3) of the pre- 
ceding article. Hence, differentiating and substituting, we de- 
rive 

d'u d'u , Q d-u sin 6 cos 6 du sin 2 

-—= — cos 2 6—2 -T-jz- + -, 

dx 2 dr drdd r dr r 

, d*u shrO du sin 6 cos ,_ 

Since the effect of putting \n — 6 in place of 6 in equations 

(1) is to interchange x and y, the expression for --^ may in this 

example be derived from equation (6) by interchanging sin 6 
and cos 6 and reversing the sign of dd. Hence 

d'u d'u . _ n d'u sin 6 cos 6 du cos 2 6 

dy dr drdu r dr r 

d'u cos 2 du sin 6 cos 8 f 

+ W ~lr~~ ~ 2 l6 r " * ' (7) 



• (8) 



44-6 CHANGE OF THE INDEPENDENT VARIABLE. [Art. 42 1, 

Adding (6) and (7) we have 

d~u d^n __ dhi I du I d^u 

~d? + ~df~d? + r dr~ + 7 dd 2 ' ' * 

422. To transform the expression 

<^ 2 ^ <^ 2 ^ ^ 2 & 
^' 2 <^/ 2 dfe 2 ' 
having given 

x = p sin 6 cos 0, 7 = p sin sin $ z — p cos 0. 

Regarding x, y, and # as the rectangular coordinates of a 
point, if we put r — p sin 6, we shall have 

x — r cos ^ and / = rsin^; 

that is r and (f>, will be the polar coordinates of the projection 
of the point upon the plane xy. r and z may therefore be re- 
garded as rectangular coordinates of the given point in a plane 
passing through it and containing the axis of z ; hence the 
polar coordinates of the point in this plane are p and 0. 
Now by equation (8) Art. 421, we have 



d*u d~u 


d*u 1 du 1 dhi 


dx 1 dy* 


dr 1 r dr r 1 d^ ' 


. d*u d*u 

dz 1 dr* 


d~u 1 du 1 dHc m 


hence, adding 





d n 'u d*u d 2 u _ d*u 1 du 1 du I d*u 1 d*u 

d? + dy 2 + 'd? ~dp Y+ ^dp + 7dr~ + '?W 2+ 7^' 

Since r takes the place of y when p and are the polar coordi- 
nates, equation (5) Art. 420, gives 

du du . n du cos 6 

~J~ — -J~ sin # + ~Jn ' 

dr dp du p 



§ XL.] EXPRESSIONS INVOLVING PARTIAL DERIVATIVES, 447 

Whence, substituting and eliminating r, we have 

d'u d*u d : u. __ ,/*/, ^2 <& cot Odu ^ I ^*« I d*u 
dx* df ' dz : " d,r pdp ir a in 8 dfi + p*~d6 i ' 

423. Examples sometimes arise in which it is known at the 
outset that u can be expressed as a function of a single variable 
which is a function of the original variables. Thus, when u is 
known to be a function of r, and 

r* = x* + JT + 2*, (i) 

let it be required to transform the expression 

d'-u d'u d'u 
~dx T ~df + dz 1 ' 

From (i) we have 

dr x dr v . dr z 

— = -, — - = - t and __=._; 

ax r ay r dz r 

, du du x du die y . dn du z 
whence -7- = -, , -j- ■ = -y • - , and — = -, , 

dx dr r dy dr r dz dr r 

d' 2 u _x d'u dr , du\~l x dr^\ _ d*u x' 1 y du r — x 2 



dx : r dr dx dr _r r dx_\ dr 1 r dr r* 

d'u _ d'u y~ du r — y' 2 . d~u __ d : u z : , du r — 

dy 1 dr r dr r dz* dr r ' dr r 3 

Adding and reducing, we have 

d'u d'u d 2 u _ d*u du 2 
dx 1 dy dz 1 dr* dr r 



44 8 CHANGE OF THE INDEPENDENT VARIABLE. [Ex. XL. 

Examples XL. 

i. Change the independent variable from x to z in the equation 

%?y dy . « 

* -r-j + ^^-+7=0, when x= e: 

d*y 
d? +y== °' 

2. Change the independent variable from .r toy in the equation 



dx> ^ y \dx. 



,2 + ^l^-J =0. 






3. Change the independent variable from y to # in the equation 

t n ^ du . s _ 

(1 — y ) — r — y — + « « = o, when y = sin x. 

x y dy f -* dy 

% d*u 

dx 1 



4. Change the independent variable from x to z in the equation 

od*y . dy t a? , 1 

x* -ri, + 2x -£- + — y = o, when a: = — . 
tfjc" ax x z 



5. Change the independent variable from x to z in the equation 

! ^ + 3 * ? " > + * fx +y = °' WhCn Z = l ° gX ' 

d 2 y 



§ XL.] 



EXAMPLES. 



449 



6. Change the independent variable from x to t in the equation 

cPx i ,/r 

,/.v= + .v^ + - , ' = ' Whe " V=4/ - 

d'y dy 

7. Given x = / + t'\ transform -,-, into an expression in which x 



is the independent variable. 



d^u . N d~u du 

-z-r = (1 + 4X) — ; + 2 



8. Change the independent variable from x to z in the equation 



/ , n //^ , dy 

(1 + x ) -7-7, + x — + 2ax — o, when 
dx- dx 



z = log [x + 4/ (1 + jr)]. 



9. Change the independent variable from x to s in the equation 



/ s s\ ^l v tf2 dy x ~ _ 

dx'- x dx a ' 



when xr + z 2 = a 2 . 

d'y 

a — - -r 1 = o. 



10. Change the independent variable from z to x in the equation 

s d JZ + z ±} when 






dz 1 dz 



Z — € 



dy dy 

-~, 4- tan x -y- 
ax~ dx 



1 = 0. 



11. Change the independent variable from x to t in the equation 



dy z 11 

— Z. 4-2 

dx 2 a' 2x + e- 2x dx ' (f'-' x + £"' x )- 



£ 2x tfV 4>6 ; 



o, when x = log V tan /. 



dy 



dt 



- + «-y = o. 



45° CHANGE OF THE INDEPENDENT VARIABLE. [Ex. XL. 

12. Change the independent variable from y to x in the equation 

d 3 u d^u „ du ... 

-j-3 — 4 tan jy -j-j 4- 2 tan'jy — = o, having given tan jy = .*. 

(i + AT) 77 + 2* (i + #') -3-7 + 2 — - = O. 

dx dx~ dx 

13. Change the independent variable from y to x in the equation 

(1 -/) -j - 2 j/(i -/)- + —— «-o, when y = 



df J v J ' dy 1 -y ' ^ £* + r 

14. Given a: = £ e , transform the equation 



d*u 
Employ dy — (1 — y*) dx. jj + a (f 2x + 1) = o 



x 3 ~- % + ax' -A + fa-j- -\- cy — o. 
dx dx dx 



Employ equation (4) Art. 417 



^ + («- 3) -33 + (»-■« + »)± + 0-= °- 



15. Show that when a — 3, b — — 2, and c = 2, the equation given 
in the preceding example is equivalent to 



r-- 



~d 



y=o. 



16. Given x = a (1 — cos /) and_y = a (fit + sin/), prove that 

d*y _ n cost + 1 
dx* ~ a sin 3 1 

17. Given # = r cos 6 and y = r sin 0, y being a function of x, 
prove that 

£y + 2 We/ ^ 

^■ 2 / dr 

cos — rsmfi 

</0 



§ XL.] 



\MPLES. 



451 



,8G,ven **?-j(£) + £=°> "> 



x=yr f 



prove that 






19. Given ^ = sin9 and y=£- } prove that 

~n = — rr 13 sin cos — sin — 2 . 
dfcr cos J 

20. When a: = £ e prove by means of equation (3), Art. 417, that 



a. d \ 
X* —-„ X* • U = 

dx 



de 



1 

S 4 



and verify when u = sin x. 
21. When 3; — s e , show that 



^ 3 d b 
dx dx 



LM + s J 



d 

7o + 1 



*/0 



22. By means of the principle deduced in Art. 418, simplify the 
expression 



b-a>i'-n 



Solution : 



r _, d; 

dx_ 



dx 



_, ^/ _ } d z d _ x d _j d 
dx dx dx dx dx ' 



in which the factors between the periods are commutative ; hence the 
expression becomes 

d^_ _, d_ a _^_ , ^__ <£_ -x.d_ d_ -x £__.£_ 

dx* dx dx " dx dx~ dx dx dx dx 5 ' 



45 2 CHANGE OF THE INDEPENDENT VARIABLE. [Ex. XL. 



~- r 2 i d 3 d 2 d 

23. Given # = £ prove that 4 -7- x — xu = x ——: u. 

dx dx dx 



Solution ; 
In this case, 

hence, by Art. 418, 



— - u == 2xu ; ( 1 ) 



d % 



dx dx 



therefore, by equation (1) 



A — X —XU=2 ~^—X -r-„ U = 2X -7-5 XU 



^c dfor 



^ 



</ 4 



4 — JP -7- XU = X -7-4 &. 

dx dx dx 



24. Prove that 

Solution : 
When * = e\ 



~d_~ 
dx 



dx 



n y=U 



dx 



r d n y 
dx n 



\ x - n — \ — n ; 

L dx [_de 



hence the symbol in the left hand member is commutative with sym- 
bols of the form considered in Art 418 ; therefore 



~d_~ 
dx 



" d "I 


r 


x — n 


= x~ n 


dx 





dx 



dx 



— K-J- 



£]'■ 



which, by Art. 410, reduces to 

25. Given x — Vz prove that 



~d_ 

dx 

~±£ 

x dx 



dz r ' 



§ XL.] EXAMPLES. 

and thence find the value of 

B£P^ andof [IsT 108 * 

See Ex. XII. i8, and Art. S4. 

2" [x* + ;;) e''\ and (* — 1) (*— 2) - - - 1 (- 2 >•"' ■*'* 

26. Given x = - , prove that • .v s — - , = — — ; 

L <frJ L < :: ^ 

and thence find the value of 

a- — - lo J .v. and of '— sm-. 

L ^ v J L ^J - v 

1 - 2-3 (* — 1) *", and (— 1 )" sin — tt ■■ 

27. By putting x = tfz in the formula proved in example 22, derive 
the equation 

and verify for the function s*. 

: : Given * = r cos 9, and;. = r sin 5, prove that 

du du du . . du du du 

x- p — = — , and that x y-^- = r—. 

dy dx in ax ay dr 

29. If £ = x cos a' — y sin or, and 7 = x sin a- + _j> cos a-, prove that 

d'u d i u _ d i u <Tu 
dx* d£' di 

30. Transform the expression r c — — — into a function in which 



454 CHANGE OF THE INDEPENDENT VARIABLE. [Ex. XL. 

x and y are the independent variables, having given x = r cos and 
y = r sin 0. 









^ </0 2 v ^ '■ [_dx* df j ^ ^ ^ 

31. If j = s x + £ y , and / = s~ x + s ~\ prove that 

c£ l u d*u d^u 9 d*u , d^u 9 d*u du du 

-— + 2 -— — -f — T = f — - — 2tf — — + r -^ + s — + /— . 
dx dx dy dy ds ds dt dt ds dt 

32. If x = as cos <f>, and y = as sin $, prove that 

2 d' 2 u d^u s ;/ 2 z/ _ « 2 « //& 

^ ^? "~ 2 ^^ + x ' d/ " ^ 2 + S& ' 



XLI. 

Lagrange s Theorem. 

424. Let j be an implicit function of the independent vari- 
ables x and z, satisfying the relation 

y = z + x<l>{y)> (1) 

in which i> denotes any function ; then, if we have 

u=f{y\ • • • (2) 

71 will also be a function of x and z. 

If now it be required to develop u in a series involving 
powers of x } we obtain by the application of Maclaurin's theo- 
rem 

du ~| • d^u 

u = Mo + —r- x + -j-. : 

ax J ax- 



f- + ---, • • . (3) 



in which the coefficients are functions of z. We proceed to 



§ XLL] LAGRANGE'S THEOREM. 455 

r ^i i • du d~u . , . , 

transform the derivatives ,, etc., into expressions in which 

Z is the independent variable, before determining their values 
when x = o. 

From equation (2) we have du — f (y) dy, hertce by differen- 
tiating equation (1) with reference to x and z, we obtain the 
partial derivatives 

du _ f\y)<i>{y) , du_ f'(v) _. ,. 

dx - 1 - ,rf(j/) ' A " I - *^0f) ' " " W 

whence Z = * ( '>sf (5) 

In order to deduce the required expressions for the higher 
derivatives, we first establish the general theorem — that, when 
y is a function of x and z y and 21 and ip (y) are any functions of 
y, we have 

s*w£=£*«£ < 6 > 

To prove this theorem, we have only to perform the differ- 
entiations ; thus, putting f {y) for u, both members of (6) re- 
duce to 

dn 
Substituting in the general theorem (6) the value of -7- given 

in equation (5), we have 

s^a-a'^^s-/ • • • (7) 

Applying the symbol — to equation (5), and reducing the 
second member by means of equation (7), we find 

dx ! dz Ln/;J dz w 



45 6 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 424. 
Again, applying — to equation (8), we have 

d?u d d r , , ... 9 du d d r , , s -. 9 du 



and, reducing by equation (7) 



d s u d 2 r , f ^«du 
by successive repetitions of this process we obtain in general, 

^=~i*U)r^- (9) 

ax dz dz 

In determining the values which these derivatives assume 

when x = o, we notice that when x = o equation (1) gives 

du 
y — z\ hence u Q =f(z), and from (4), -7- =f'(z). More- 

az _J x — 

over, since the differentiations indicated in the second member 
of equation (9) have reference only to z; we may, in this equa- 
tion, assign to x its value before the differentiations are effected : 
therefore 

Substituting these values in equation (3), we obtain 

f(y) =/(*) + **«/>) + f^i ) W (*)]*/'(*) (+ • • • 

1.2 •• -n dz ( ) 

This result is known as Lagrange's Theorem. 



§ XLL] LAGRANGES THEOREM. 457 



425. As an application we expand the function 

y = z + xe y . 
In this example f(y) = y .-. /' (j) — i, and ^(^) = r i 
The general term is therefore 

n jn - i , : 

*** = *— ' s MS - • 



1-2 • • • ;/ dz n ~ l 1-2 

whence 

r 2 r 

y = z + £* • X + 2 f 2 ~ • h 3 2 £ r 



2 I- 2- 3 

To obtain the development when the function is 

y = I + .r^, 

we put I in place of # in the preceding development. (Com- 
pare Ex. XXIII, 14.) 

426. When the given relation between x and y is not in the 
form required for the application of Lagrange's theorem, an 
algebraic transformation sometimes enables us to make the 
application. Thus, if we have 

\ogy=xy (1) 

to develop y in powers of x, we put log y — y' ; whence we have 

y = e 3 , and (1) becomes y' = xe y . 

The latter equation being in the required form, we have 

u—y^ e y \ f (/) — s y \ and <j> (/) = s y '. 

Hence, the general term is 

s - \n + I) e , 



I-2- • • n dz l J 12 



45 8 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 426. 
and putting z = O, we have 

^ ° 1-2 1-2-3 

427. Lagrange's theorem may, in fact, be applied to f (y) 
whenever the relation between y and x is of the form 

y = F [z + x<f) ( y) ] ; 
for, if we put 

t = z + x<l> (y), we have y = F (/); 

whence « =fF{t), and / = # + x<f)F (f). 

Lagrange's theorem is therefore immediately applicable, 
the functions fF and (j)F taking the place of f and <f> in the 
development. Hence, substituting, we have 

This form of the series is called Laplace s Theorem. 

428. The example in Art. 426 may be regarded as a case of 
this theorem ; for the given equation may be written in the 
form 



and we have in this case/(j/) = y, also 

F (t) ■=£*, t — xy, z — O, and <f> (y) = y. 

Both fF(z) and $F (z) reduce to e z , and are identical with/(V) 
and <f> (z) of Art. 426. 

Since Lagrange's theorem is simply Maclaurin's theorem 
with transformed coefficients, it is always possible to make any 
series derived by its application convergent by giving to x a 
value sufficiently small. See Art. 157. 



§ XLI.j DEVELOPMENT IX SERIES. 459 

The Development of Finn/ions of Two Independent 

J 'aria dies. 

429. Let *=/(*,?), (0 

and let it be required to develop f{.v + h, y + k) in a series 
involving powers and products of powers of h and k. Let r de- 
note any assumed interval of time, and put 

a=-, and b — - (2) 

x r 

• If now we assume 

x — x +at and y=y Q +bt> ... (3) 

/ denoting a variable interval of time, u will become a function 
of /, and we may write 

u=f(x +at, y +bt) =*(/) . . . . (4) 

• Putting t = x in this equation, we have by equations (2) 

Developing <f> (r) by Maclaurin's theorem, we have 



(t>(r) = 0(o) + rf (o) + — f'(o) + • • ■ 

+ — f{o) + ^ v ^ + I (0r). . . (5) 

1-2 • • •« W 1-2 •••(»+ i) V ' V:?/ 

Since ^ (/) = » is a function of ;r and y, we have 

,, . s _ <^?/ _ du dx dit dy 
( ' ~ It ~ dx dt + dy df 9 



460 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 429. 

or, since by equation (3) ~ = a and ~- = b, 

at dt 



f (0 = a 


du j du 

dx dy 




xi x" /*\ d~u „ d°'u 7 d 9 u .„ d*u 
Hence «5 W -^ = "° ^ + 2 "*^ + * V 


and in general 


7« 
f« /A ^ 2/ 


d , d 
a — + b — 

- dx dy~ 


n 
U 



(6) 



Putting / = o in the expressions for $ {t), <j)' (t), etc., and sub- 
stituting in (5) we have 



du 



du 



f(*o + h > )'o + k) =f(x Q , y ) + r\a—+_ 

2 d*u 



r 

+ — 

1-2 



d'u 
'aW 



+ 2ab 



dh 



+ b 



dx dy dy 2, J 



+ • • • + R. 



Substituting for a and b their values from equation (2), and 
omitting the subscripts since x Q and y Q alone appear in the re- 
sult, we have 



/(* + h, y + k) =f(x, y) + h *? + k p. + 

ax ay 



I r j d , d 

— h -3- + k -J- 

i-2[_dx dy _ 



u "+ • • • + R. . . (7) 



In equation (5) the remainder is obtained by putting / = 6r y 
instead of t — o, in the expression for the (n + i)//^ deriva- 
tive, and since t — Or in equation (3) gives x = x Q -f Bh and 
y = j/ 4- #/£, we have 



tf = 



r* ^ 






1-2 •• • (;z + t)[_ dx dy 






r + 0^, 7 + 0£). 



g XLL] THE SYMBOLIC EXPRESSION FOR THE SERIES. 461 



The Symbolic Expression for the Scries. 

430. Since the coefficients and indices in the above result 
follow the law of the exponential series, equation (7) may be 
written in the symbolic form 

f(x + /i, y + k) = e d * + <* fix, y). 
It follows from Art. 176 that the application of the symbol 

€ dx to u is equivalent to changing x to x + h. Accordingly 

, ^_ 
the application of the symbol e "> to the result thus obtained 

is equivalent to changing y to y + k. The preceding demon- 

u — + k - 
stration shows therefore that the symbol £ dx ' d > is equiva- 

h— k — 
lent to e **•€&, and the form of the series shows that the 

h— k- 

symbols f dx and a d >' are commutative. 



In like manner, were u a function of .?, as well as of x and y, 

d 
the application of the symbol 8 dz to the result obtained above 
would change ,z to z 4- / ; whence it may be inferred that 

/— A— + k — 

f (x + k, y + k, z + I) = a ** e d * + d > f{x, y, 2) 

d_ d_ d_ 
= e** + ** *f(x,y,B). 



Maxima and Minima of F mictions of Two 
Indcpcn den t Va riablcs. 

431. A function u of two independent variables x and y is 
said to have a maximum value corresponding to certain values, 
a and &, of x and y, when after increasing, u begins to decrease, 
as x and y pass simultaneously through these values, whatever 



462 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 43 1. 

be the comparative rates with which x and y vary. A minimum 
value of u is defined in a similar manner. 

It is obvious that a maximum value, as defined above, must 
continue to be a maximum when either y or x is made con- 
stant ; hence both -=- and -=- must change sign from + to — , 
ax ay & » 

as x increases through the value a and y increases through the 
valued. See Art. 119. In .like manner for a minimum both 
derivatives must change sign from — to +. 

The derivatives can change sign only on passing through 
infinity or zero ; it is only when both derivatives pass through 
zero that criteria founded upon the higher derivatives can be 
employed for discriminating between maxima and minima. 
This, however, is the case which most frequently presents itself. 

432. Let a and b denote values of x and y determined by 
the simultaneous equations 

du . du ,. s 

Tx =0, and ^ = (I) 

If we cause x and y to pass through the values a and b with 
the arbitrary rates 

• £=* - d %-' & 

u will become a function of t. Hence we shall have , , 

du _ du dx du dy __ du „du % , , 

dt dx dt dy dt dx dy 

du 
therefore, when x = a and y — b, -r- is zero for all values of a 

and (3, Hence, by Art. 133, we shall have a maximum value 

, , d'u . . , . . , , d*u . 

of u when -— is negative, and a minimum value when -^- is 



§ XL I.] MAXIMA AND MIX IMA. 463 

positive. Differentiating equation (3), the arbitrary rates a and 

ft being assumed to be constant, we have 

<Pu . (PU d*U ,,.,d U , s 

= a -7-5 + 2a'/i ,,- + /; — . . . . (a) 

df 1 dx 1 dxdy <iy ™ 

Putting A=- r7 \ £ = - and C = , 

we shall therefore have a maximum or a minimum when the 
expression 

A a 9 + 2Bafi 4- Q? (5) 

retains the same signs for all values of a and ft. Writing this 
expression in the equivalent form 

£[(C£ + Bf+{AC-Jri], .... (6) 

it is obvious that this condition will be fulfilled when we have 

AC -B 2 >o. 

This is known as Lagrange s condition. When this condition 
is fulfilled, A and C have like signs, and the sign of the expres- 
sion for -=-y is, for all values of the ratio - , the same as that of 
dt ol 

A and C. Hence the value of u will be a minimum when A 
and C art positive, and will be a maximum when A and C are 
negative. 

433. As an illustration, let the function be 

u — a- -r b- — x* — y- — 2b s' {ci- — y") ; 

, du , du 2by 

whence — — = — 2x and -=— = — 2v - 



dx dy V{a 2 —/) 



464 FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 433. 



These derivatives 


vanish 


when 


x = 


and y — 


0. 


We 


have 


also 
















d' 2 u 
dx^~~ 2y 


d*u 

dx dy 


= 0, 


d*u 

~df 


— — 2 + 2 




be? 




(«■ 


~ff 


y 


therefore, when y - 


= 0, 














A=-2 


B 


= 




C= -2 


+ 


b 
2 a< 




whence 


AC- 


& = i 


a — 
\ 


b 









Hence, ita>b, Lagrange's condition is satisfied, and u is a 
maximum for x = o, y = o, since A and C are negatived If, on 
the other hand, a < b } Lagrange's condition is not fulfilled, since 
AC — B 2 is negative, consequently there is neither a maximum 
nor a minimum. In this case, expression (5) becomes 



- 2 p-^-V], 



which is negative when — 2 <- , and positive when •— > 



or b — a or b — a. 



434. When A — o, B — o, and C — o, the expression for — 



d'u 
~di> 

vanishes for all values of a and /?; hence (see Art. 138), it be- 
comes necessary to examine the higher derivatives. 
The general form of the m\h derivative is 

d m u f d ■ R d\ m , , 

a— + fi-j-) u (I) 



JtZ \ dx dyj 

This expression when expanded involves all the partial deriva- 
tives of the ;/zth order with reference to x and y ; hence it will" 
vanish for all values of a and (3 only when all the partial deriv- 
atives of the mth. order vanish. Now let n denote the order of 



§XLI.] MAXIMA AND MINIMA. 465 

the lowest derivative that docs not vanish. It is shown in Art. 
138 that there can be a maximum or a minimum only when 11 is 
an even number, and it is further necessary that the expression 

d"u 
for - shall retain the same sign for all values of a and 6, This 

df 

will be the case only when all the values of — derived from the 

a 

equation formed by putting the expanded expression for this 

derivative equal to zero are impossible. 

435. If we have AC—B'- = o when A, B, and C do not 
vanish, expression (6), Art. 432, has the same sign as C for all 
values of a and (3, except those which make 

C P- +B = o- that is, £ =-|L 
a a C 

When this case presents itself it is necessary to examine the 
higher derivatives for this particular value of the ratio. The 
illustrative example in Art. 433 furnishes an instance of this 
case. For, if a — b, the expression for the second derivative is 
— 2a- indicating a maximum except when a = o. Now when 
= o equation (1) of Art. 434 becomes 



a 



d" l U _ d'"7C 

~dT ~ dy m ' 

and from the value of -=- already determined we derive 
df 

d^U s-1 o / o o\ 5. dti ri *V( « 2N- 5 - , - o / "\-"-| 

df = ^ ^ "^ ' > df = ^ '^ ' SJ " {a '~ y ^ ^ 

Hence, when y — o, the third derivative vanishes and the 
fourth derivative becomes 6bffa~ 3 , a positive quantity, indi- 
cating the existence of a minimum when a = o; but since we 
have a maximum for all other values of a it is obvious that the 
value of the function is not in this case a true maximum. 



FUNCTIONS OF TWO OR MORE VARIABLES. [Art. 436. 

du du 

4-36. When — - and —=- have a common factor, by putting 
ax ay 

this factor equal to zero, we obtain a relation between x and y, 
and it is obvious that these derivatives will vanish for all values 
of x and y which satisfy this relation. Such values do not how- 
even correspond to true maxima or minima values of u. For, 

du 
when x and y vary, subject to this relation, it is evident that — 

ai 

will remain zero, and consequently that u will remain constant ; 

that is, it is possible for^r and y to vary so as neither to increase 

nor diminish the value of u, therefore u is neither a maximum 

nor a minimum. As an illustration of this case, see examples 

13 and 15 below. 



Examples XLI. 

1. Given y = z + xe p \ expand e my in powers of x. 



1-2 



1-2 • • '11 



in {iip + m) n ~ 1 &*+* 



2. Given y = a + xy\ expand y in powers of x. 

X' 

y = a + ax + 6a 5 — + 9 - 8a 1 - 

y 12 1-2- 



X 



X 

+ 12-u-io a 9 h 

1. 2-3-4 



3. Given y = 2 + x , expand y in powers of x. 



z + 1 (c 2 - 1) x + Js (3 2 - 1) x- + frfe* 1 ~ 6 **+ x ) ** + 



§ XLL] EXAMPLES. 467 



4. Given x = - 4 x sin r, to expand cos v in powers <-t A. 

4 

1 .v 3# 9 a; 3 a" </""' , . 

cosj' = — * ...•—. — - sm ; ) 

V 2 2 4 , v / 2 3 1 • 2 • • • n (/:■'• ~ l v ' 

5. Given y = a 4- />r u , expand j- in powers of />. 

y = a + tf"£ 4- 2Jia'"~ \- t« (*« — 1) d v '~' \- • 

J 1-2 ^ VJ ' 1-2-3 

6. Given ^ 3 + \x + 2 = o, determine the value of x. 

, = _*+' J,. +£_. 

2 2 2 

7. Given y = a + xf, expand y s in powers of x. 

^x 2 1 v s 

f =z a 3 + $a a x 4- 8« 7 - — 4- n-ioa: 9 -* — 4 ■ - 
1 2 1-2-3 

8. Given y = £ 4 -v logj 1 , expand/ in powers of x. 



2 A" 3 x a. x . 

y = S + X+ • 4" -a ' ~ - — "a " * ' 

€ 1-2 £ 1-2-3 6 12 ' " '4 

9. Given _>> = xs~ v , expand sin (a 4 y) in powers of x. 
Sohction : 

x n . d"' 1 

The coefficient of is -7-— \£~ nz cos (a' + z) 1 ; 

1-2 ■ • • n dz"' 1 L v J 

and by equation (2), Art. 410, we have 

—- £ cos (a + z) = € "~ [ — — n cos (a 4- s). 

Now f « J cos (a + z) — — sin (a 4* z) — n cos (<? 4- z) } 



46S FUNCTIONS OF TWO OR MORE VARIABLES. [Ex. XLL 
or, putting n = cot 6, 

( - 7 11) cos (a + z) = : — - cos (a + z — Q). 

\dz J v 7 sine v J 

In like manner, 



and finally, 



* V i 

11) = -t-=- cos (a: + 2 — 20), 

afe / sure v 



— — n J cos (or + e) = ( — i) n_I (cosec e) tt_I cos [a + z — («— i) 6] 

Putting for cosec Q its value Y(i + «' 2 ), and putting z = o, we have 
for the general term 

■ (- i)- 1 (i + ** 2 )~ f" ws cos [a - Cn — i) cot "Vl 

i-2 • • • n 

.*. sin (# -f- jv) = sin a + x cos «+'••• 



— (— i) n (i + n) 2 cos \a — (/j — i) cot" 1 //] + 

i • 2 • • • 11 



io. Develop [i + j/(i — /)] in powers of e. 

Put E — i + 4/(1 — **), whence J? = 2 — - 

Determine the maxima and minima of the following functions : 

11. u = x* + / — $axy. A min. for x — a and y — « : 

neither for x = o and jy = o. 

12. « = Jf 4 + J/ 4 — 2JV 2 + 4JCJ/ — 2j)/ 2 . 

A min. for x — ± 4/2 and jy = =F V2. 
neither for # = o and y = o. See Art. 435. 



§ XLL] EXAMPLES. 469 

13. // — <7x v~— x*y* — -vi -f 0*. A max. when x = antl v = - . 

3 

The case considered in Art. 436 arises when x = o, also w hen r = o. 

, * 3 a v . . a 

14. // = x + at + v" H 1 . A mm. when x = y = 

.v j- 

2 2 

15. » = £" a "•"' (^v- - 1 - by'). A min. when .r = o and y — o. 

I{ a > b, neither, when x = o and v = ± 1 ; 
a max., when x= ± 1 and r = o. 
If a — b, the case considered in Art. 436 arises when x' J + r = 1. 



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